To be more clearer with the question let us compare for this instance sociology to philosophy, both being a broad subject. Sociology has many branches including political science, psychology, anthropology, etc., yet political science can be discussed without the participation of anthropology in other words it can exist by itself, since their objectives are compleltely distinct though sharing the same referent, the society. In philosophy most branches has the suffix "philsophy" to it like political philosophy, moral philsophy, relgious philsophy, etc., however they have different goals though sharing the same philosophical method. So is a philosophical branch an independent subject by itself than just being a branch under one subject or in other words can a certain branch of philosophy exist independently from other branches as in sociology?
1) Yes. Philosophy isn't holistic (or necessarily so). Until recently, philosophy itself wasn't an independent field of study. Remember that physics used to be called "natural philosophy." Newton's Philosophiae Naturalis Principia Mathematica translates as the "Mathematical Principles of Natural Philosophy." Philosophy, as an entity, is difficult to nail down: there are philosophy of mathematics and philosophy of science courses at most universities. Yet, what is the difference between an course on electromagnetism and a course on the philosophy of physics (or even the difference between the philosophy of physics and the history of physics)? --Perhaps, the difference is problem solving....
But what the hell is this "philosophy?" Is it so enigmatic that it continually changes its shape? Was it once the "love of wisdom" and something else now? Aristotle thought that wonder paved the way for philosophizing:
For it is owing to their wonder that men both now begin and at first began to philosophize; they wondered originally at the obvious difficulties, then advanced little by little and stated difficulties about the greater matters, e.g. about the phenomena of the moon and those of the sun and of the stars, and about the genesis of the universe. And a man who is puzzled and wonders thinks himself ignorant (whence even the lover of myth is in a sense a lover of Wisdom, for the myth is composed of wonders); therefore since they philosophized order to escape from ignorance, evidently they were pursuing science in order to know, and not for any utilitarian end. And this is confirmed by the facts; for it was when almost all the necessities of life and the things that make for comfort and recreation had been secured, that such knowledge began to be sought. Evidently then we do not seek it for the sake of any other advantage; but as the man is free, we say, who exists for his own sake and not for another's, so we pursue this as the only free science, for it alone exists for its own sake. (From Aristotle's Metaphysics, Book I, trans. Ross)
If the above account still applies to philosophy today, then it would seem that, as a whole, philosophy encompasses quite a bit; and yet, its branches are, or can be, quite separate. Few people today would claim that thermodynamics is a philosophical discipline; but, for example, Maxwell's demon is not an experiment for physics: it's a thought experiment. You see, philosophy isn't like other disciplines (I, for one, don't consider it a subject at all), and in many ways, philosophical dilemmas and questions helped create most, if not all, of the disciplines people study today. But, in my long, and perhaps fittingly rambly and scattered response, you should see that philosophy is not the kind of thing you thought it was.
2) Yes. Many philosophers only concern themselves with one or a few of the so-called branches of philosophy. For example, Wittgenstein was mainly concerned with logic, the philosophy of language, epistemology, and metaphysics; he didn't write about political philosophy or ethics (except for one, short lecture on ethics).
No, the branches of philosphy can´t exist independently. I would even claim that if we think of another possible world where there is only political philosophy, the other branches would soon turn up as well.
Why is that?
Let us think of our political philosopher, sitting in his garden. In order to do his work, he needs to know that there are other things besides him in general, e.g. other humans (ontology) and how we get to know about them (epistemology), that other humans have a consciousness (philosophy of mind), basic concpets of what is right and what is wrong (ethics) and how to properly speak and form rational arguments (philosophy of language and logic).
Of course one could argue that logic (not as a branche, but as logic itself) infact exists independently insofar as there would be the rules of logic even without humans and other living things. To answer this you would have to do some ontology, though ;)
Which then leads me to the question: Where is the border between science and philosphy (if there is one) and could science exist without philosphy?
One border might be the existence of experiments in science and the lack of it in philosphy. The latter one can be answered at least historical. Most of our sciences emerged out of philosophy. Still, the same motivation that attracts people to the sciences would motivate them to do philosophy aswell.