If by "assume," you mean "accept without argument," then the answer is yes to the first part of your question and no to the second.
Yes: philosophers assume all sorts of things. They usually assume that there is a world out there and that there are people who at least potentially can read and respond to their arguments. They very often take for granted all sorts of facts, scientific and garden-variety: that there are trees; that people sometimes do things deliberately; that water is made of H2O; that the 4-color conjecture has been successfully proven.
No: this isn't contrary to the intent of philosophy. Except on very eccentric views, philosophy is not the enterprise of doubting everything that can be doubted and accepting only what can be proved from indubitable premises. That may have been Descartes' project, but it's been almost no one else's. In fact, most philosophers would say that this project is deeply flawed.
Philosophy, like the Odyssey begins in the middle of things. Want to think about free will? Start by assuming that there are beings who have intentions and sometimes behave accordingly. Want to think about linguistic meaning? Start by assuming that there are languages and people who use them to communicate. Want to think about cause and effect? Start by assuming that things happen (there are "events," on one way or putting it) and that sometimes what happens fits certain sorts of patterns. And so on.
But in any case, there's no one thing that is the "intent" of philosophy. True: philosophy is relatively non-empirical; we don't generally do experiments to settle philosophical questions. It's hard to see how experiments would tell us what it amounts to for one thing to cause another, for example. But sometimes empirical findings are relevant to philosophical thinking. For example: what experimental psychology tells us about how we actually make decisions might very well be relevant to how we should think about free will.
Who decides what gets doubted and what gets taken for granted? The philosopher, and she does it depending on the questions she wants to explore and the state of the broader discussion of those questions. All of this may seem distressingly impure to outsiders, but philosophers would say that it makes it possible to have productive discussions that someone might have a reason to care about.