To understand how Sartre could ever say something like this, we need to look at an important pair of German philosophers and one Dane (actually we could probably find many more important people in this transition) and their effect on how we think.
I assume by classical definition of essence with reference to Aristotle, you're thinking that each natural kind has an essence, and that defines what the thing is. Thomas Nagel makes the point vividly in talking about giraffes in Amelie Rorty (ed.) Essays on Aristotle's Ethics. And Aristotle derives his account of human ethics in Nicomachean Ethics in part from this idea.
So far so good. But as time passes, the word essence (which of course can't be Aristotle's word since it's latinate) gets tied up in a debate about whether or not essences follow something like Plato's ideas, Aristotle's "that what it is to be" --> essence, or something that comes from our heads. (Moving through the middle ages at a very rapid clip, we have Augustine sounding Platonic about this, Aquinas sounding Aristotelian, and Ockham and Buridan sounding very "in our heads" about essences).
Then we get to modernity. The rationalists following Descartes and then Kant present what is largely a dualistic account where what it means to be human has little to do with the embodied essence that Aristotle suggests and much to do with our use of reason. Looking just at Descartes, we could say that the human essence is rationality. And For Kant (in his ethics), humanity and rationality are synonyms.
This takes us to Hegel who makes an interesting claim that the fundamental nature of humans is not just reason but also will. In other words, what makes us human is that we impose categories and distinctions on the world we encounter (and generate?).
Kierkegaard, too, emphasizes the sense in which humans are beings that relate differently to the world. In the beginning of Sickness unto Death, he suggests that what makes us different from animals is that we can despair and that we have choices about how we relate to the world.
Sartre is largely building on Hegel and Kierkegaard (there are strong reasons to believe he had read both). And the idea he's proposing is that rather than having a set pattern of life like a giraffe, we have the power to choose how we set our lives. This is what he means by "essence" and the suggestion that humans are in a unique position to pick their essence.
I will try to make that more practical.
A squirrel does not have a choice in figuring out how to live squirrely. It does not (as far as I can tell) ever have to decide "well, what does it mean to be a squirrel? what are the interests that squirrels ought to have? how well am I doing in achieving squirrelish excellence?" All of these question are truncated in the immediate mission to collect and bury nuts.
Conversely, humans do have a choice about their vision of ideal humanity. Thus, we can debate whether it makes sense to be monogamous and form marriages, to be heterosexual, to be obese or at an "ideal weight," to pursue higher goods and intellectual achievements, to pursue justice. In other words, I could decide my ideal life is just like the squirrels and try to live that out -- living in a tree, hopping around, collecting nuts, eating them raw with my teeth. Both other squirrels and other humans might find it odd, but it's within my power to try to choose my essence.
This is not to say that there cannot be answer, but in part that we have a further task of rightly picking that answer (it also does not rule out that there is no answer). (The squirrelish life doesn't seem well suited to us, because we don't have teeth that regenerate, there aren't tree holes the right size for us, etc. etc.)
tl;dr - when Sartre says we pick our "essence", we can reword that as him believing we have a choice in selecting what we believe our essence to be. And in the absence of this choice, we don't have an essence that we know about ourselves.
The above serves as an answer to your question. You can source a lot of it to Existenialism is a Humanism, but you can find a much more robust and difficult to read account that handles objections better in Being and Nothingness. The latter account is largely parallel to Hegel's but skeptical in places where Hegel is hopeful about our capacity for really comprehending things.
Regarding question 2, I think Sartre is using the term in a way that's different from Aristotle and by reason of the history of the term doesn't find himself to be committed to a contradiction. In large part, this is because the Hegelian frame emphasizes that we are the ones putting categories that we decide into the world.
Regarding questions 3 and 4, Sartre's argument (granting it the most leeway) would apply to creatures who create and manage their own concepts of world. This needn't just be humans, but it would not apply to the sort of animals and plants that don't frame and dice their world according to their own concepts. To use, Heidegger's language, many animals are world-poor. Sticking with our squirrel, he doesn't ever seem to contemplate whether life is all about nuts and chasing female squirrels. That's just life, so his essence is set. (at least as far as I can perceive). If there's a being that asks, "what am I here for?" and can make choices about that, then the "existence precedes essence" dictum would apply.
