No. You are free to choose what you want to use. But what's the point on using free software and using non-free fonts? Free software is not about software any more, the whole point now is free culture.
Not if you don't want to. But there are enough free fonts on openfontlibrary and elsewhere to do just about anything
You do not have to, but why not?
It might. There are lots of good ones. But you can also use the proprietary ones.