I am not exactly sure about these two ideas. For me, it seems that logicists believe that axioms means something more than a string of symbols, and mathematics reduces to logical facts, so they probably believe that there are truths in an eternal and unchanging “world of mathematical Forms”.
Specifically, my questions are
Logicism -- as usually treated in the literature -- is a claim of reducibility, that (to oversimplify brutally) one or more core branches of pure mathematics can be analytically reduced to more basic truths of pure logic.
Platonism -- as usually treated in the literature -- is a metaphysical claim about the ontological status of the facts or objects that explain mathematical or logical principles. It holds (to oversimplify brutally) that these are to be explained by the existence of, properties of, and relationships between real abstract objects independent of human minds, thought or practices, and also independent of, external to, and/or in some sense transcending objects in or facts about the natural world of things, time and change.
If you are a platonist about mathematics, then you think that there are at least some abstract mathematical objects (for example, numbers) that exist, and at least some mathematical truths about those objects (for example, that some numbers are prime and that there is no greatest prime number) that are true independently of human thought or human action, and eternally regardless of the physical constitution or configuration of the world around us. (This is usually combined with the epistemological claim that these objects and the relations between them are discoverable by human reasoning or intuition, and that this process of discovery is what explains successful mathematical learning.)
If you are a platonist about logic, then similarly, you hold that logical truths are explained by the existence of, and relations among, abstract logical objects that exist independently of human minds, thought or practices, and also independent of, external to, and/or in some sense transcending objects in or facts about the natural world of things, time and change. Correct logical reasoning is explained in terms of principles that correctly represent the real, preexisting relationships among logical objects.
If that's more or less what you meant by both of the key terms in your questions, then let's try some answers. If logicism is a claim about the reducibility of mathematical truths to logical truths, and platonism is a claim about the ontological status of mathematical and/or logical truths, then what logicism should tell you about mathematics is that any proposition in mathematics can be shown to be analytically equivalent to a set of propositions in pure logic, so the propositions of mathematics can be shown to have the same ontological commitments that propositions of pure logic have, whatever those may be. Platonism about logic is one particular view about what those ontological commitments are.
So IF platonism about logic is true, THEN logicism about mathematics would entail platonism about mathematics. On the other hand, if you are committed to some other theory about the ontological status of logic (for example, you could be a naturalist about logic, or a conventionalist, or constructivist, or...), then logicism about mathematics would not commit you to platonism about mathematics; it would commit you to that other theory (so you'd be a naturalist about math, or a conventionalist, or a constructivist, or...).
But IF you reject logicism about mathematics, THEN the metaphysical questions about platonism (a) for mathematics and (b) for pure logic will turn out to be orthogonal questions, rather than the answer about one (about logic) determining the answer about other (about math). So in this case you could be a platonist about logic and not about math, or you could be a platonist about math but not about logic, or you could be a platonist about both, or about neither; one just doesn't tell you about the other.
So then to answer your questions directly, and in order:
Are Logicists necessarily Platonists? No, they are not necessarily so. These are two distinguishable questions, and logicism only commits you to a conditional claim about platonism in math (if platonism about logic is true, then so is platonism about mathematics), not a categorical claim.
Can I be both a Logicist and a Platonist? Sure, yes you can. Some logicists definitely were platonists about both logic and math (Gottlob Frege, to take the most famous possible case). Other logicists were not platonists, because they were not platonists about logic (for example, you might understand the early views of Carnap and the Vienna Circle this way).
What are the distinctions between Logicism and Platonism? See above and see below. Logicism is best understood as a claim about the linkage (or lack thereof) between mathematics and logic. Platonism is a separate claim about the metaphysical status of either or both.
I said that I had brutally oversimplified a couple of points. I can't cover these all without making this both far too long and far too tendentious. But some notes to make this more sophisticated, and give you some more sources: Logicists are usually concerned with the reducibility of truths in arithmetic and real analysis to sets of logically necessary truths about purely logical objects. (For example, Frege, in the Foundations of Arithmetic, characterizes his view as the view that the truths of arithmetic could all be known through analytic apriori judgments. Kant held that both geometry and arithmetic were bodies of synthetic apriori knowledge; Frege thought he was right about mathematical geometry, but intended to show that Kant was wrong about arithmetic. In theory, you can be a logicist about some branches -- topology, group theory, whatever -- and not about others, depending on what you think you can prove from pure logic and what you think you can't. Historically, the status of arithmetic is overwhelmingly the topic that people have argued about since Frege, Russell and Whitehead.) A second complication -- I'm going to talk here about mathematical truths being provable from principles of pure logic. But some forms of logicism restrict themselves to a claim about provable mathematical truths, which involves a weaker claim.
Here's the way a common tertiary source, the Stanford Encyclopedia of Philosophy, puts it:
Logicism is a philosophical, foundational, and foundationalist doctrine that can be advanced with respect to any branch of mathematics. Traditionally, logicism has concerned itself especially with arithmetic and real analysis. [...]
Both versions of logicism—strong and weak—maintain that
All the objects forming the subject matter of those branches of mathematics are logical objects; and
Logic--in some suitably general and powerful sense that the logicist will have to define--is capable of furnishing definitions of the primitive concepts of these branches of mathematics, allowing one to derive the mathematician’s ‘first principles’ therein as results within Logic itself. (The branch of mathematics in question is thereby said to have been reduced to Logic.)
[Tennant, Neil, "Logicism and Neologicism", The Stanford Encyclopedia of Philosophy (Winter 2017 Edition), Edward N. Zalta (ed.), URL = https://plato.stanford.edu/archives/win2017/entries/logicism/]
"Platonism" is of course an extremely tricky term, since it goes back several thousand years and depending on context it might refer to very specific recent debates in metaphysics, or it might refer back to nearly the whole documented history of ancient, medieval and modern philosophy; etc. etc. I've tried to limit myself mostly to the family of doctrines that are usually described as "platonism" in modern Analytic ontology, philosophy of logic, and philosophy of mathematics, and specifically not to any kind of historical question about how best to understand Plato or Plato's Socrates or the doctrine of the Ideas or the Forms as seen in the Republic or the Phaedo or the Meno or..... Again, here is the SEP with a decent tertiary-source explanation of what the debate there is largely about when it comes to philosophy of mathematics:
Platonism about mathematics (or mathematical platonism) is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices. Just as electrons and planets exist independently of us, so do numbers and sets. And just as statements about electrons and planets are made true or false by the objects with which they are concerned and these objects’ perfectly objective properties, so are statements about numbers and sets. Mathematical truths are therefore discovered, not invented.
The most important argument for the existence of abstract mathematical objects derives from Gottlob Frege and goes as follows (Frege 1953). The language of mathematics purports to refer to and quantify over abstract mathematical objects. And a great number of mathematical theorems are true. But a sentence cannot be true unless its sub-expressions succeed in doing what they purport to do. So there exist abstract mathematical objects that these expressions refer to and quantify over.
Frege’s argument notwithstanding, philosophers have developed a variety of objections to mathematical platonism. Thus, abstract mathematical objects are claimed to be epistemologically inaccessible and metaphysically problematic. Mathematical platonism has been among the most hotly debated topics in the philosophy of mathematics over the past few decades.
[Linnebo, Øystein, "Platonism in the Philosophy of Mathematics", The Stanford Encyclopedia of Philosophy (Spring 2018 Edition), Edward N. Zalta (ed.), URL = https://plato.stanford.edu/archives/spr2018/entries/platonism-mathematics/]
And here's what it has to say about platonism in metaphysics, which deals more broadly with the topics that I've described as platonism about logic above:
Platonism is the view that there exist such things as abstract objects — where an abstract object is an object that does not exist in space or time and which is therefore entirely non-physical and non-mental. Platonism in this sense is a contemporary view. It is obviously related to the views of Plato in important ways, but it is not entirely clear that Plato endorsed this view, as it is defined here. In order to remain neutral on this question, the term ‘platonism’ is spelled with a lower-case ‘p’. (See entry on Plato.) The most important figure in the development of modern platonism is Gottlob Frege (1884, 1892, 1893–1903, 1919). The view has also been endorsed by many others, including Kurt Gödel (1964), Bertrand Russell (1912), and W.V.O. Quine (1948, 1951).
[Balaguer, Mark, "Platonism in Metaphysics", The Stanford Encyclopedia of Philosophy (Spring 2016 Edition), Edward N. Zalta (ed.), URL = https://plato.stanford.edu/archives/spr2016/entries/platonism/.]
Logicism arises with Frege after a period referred to in mathematical folklore as the crisis in geometry. This crisis led mathematicians to pursue the exactness of their science through numbers alone. This is referred to as the arithmetization of mathematics. Using the method associated with definite descriptions and inclusive disjunction, Frege had been able to "define" a zero and a conception of numeric succession upon which one could formulate a counting arithmetic.
This period had also been marked by reconsideration of Leibniz since the crisis in geometry had been seen as undermining Kant (Ewald has translated a paper demonstrating the misrepresentation of Kant on this matter). Leibniz' principle of the identity of indiscernibles is a reflection of how Leibniz had inverted the order attached to the extensional characterization of classes by an intensional order wherein genera would actually be parts of species. Frege rejected this intensional logic with his distinction between concepts and the extension of concepts. He declared that logicians should be addressing "truth" and effectively bound this idea to naive comprehension over the (presumably) well formed objects of material reality.
His zero, however, represents an extension with no material constituent.
Frege's work introduced the function concept into logic, thereby changing the emphasis from the analysis of subject/predicate forms. Russell pursued this idea fruitfully. For Russell, logicism was to serve as a ground for science. In his "Principles of Mathematics," he explicitly discusses how he is formulating his ground for mathematics so that the philosophical position of monism will be incompatible. That is, emphasis is to be placed upon the witnessing of a multiplicity of material objects and the satisfaction of propositional functions which partition that multiplicity.
Logicism is realist because its origins are metaphysical. Frege's zero is precisely defined through a denial of the law of identity.
That is not to say, however, that logicism ought to be demarcated by these origins. Russell's paradox had shown that naive comprehension is problematic. Russell's dissatisfaction with Frege's account of truth led to his description theory. And, it is clear that Russell is adjusting his views throughout the period in response to various criticisms and developments by others.
Conifold is probably correct about modern logicists being platonistic. Nevertheless, the original idea had been that extensions of concepts be grounded over material objects with further application of the method yielding higher order objects like "numbers."
If my reading of Frege and Russell is generally correct, the answer to your first two questions is "no." As for the third, do "classes" or "extensions of concepts" actually exist in the "singular"? Frege naively assumed that descriptions referring to comprehensions could be used referentially. It didn't work. And logicians still seem to have not learned from this.
