Any reason why it can't be some deep roots? I have a crazy/stupid idea of using that garden to grow something, and roots would be rather troublesome with digging and stuff.

I don't know if roots regrow the top bit, or if it's conceivable. Sorry about that. But yes, roots can be very troublesome when digging. They get in the way and their job is to stay in place. Them and rocks, if your soil is so inclined can be a bit annoying

By the way, what should I do about dead branches? There's a lot of dead branches and I'm not sure what to do with them.

It depends on how dead, basically. If they are decaying organic matter barely discernable from the soil, it could be useful for you should your soil have insufficient organic matter (and/or acidity since they often go hand in hand). Or, if it's too acid then it could be worthwhile getting rid of it. But that's only once you start looking at the soil quality stuff.

More importantly, if those branches are making it hard for you to access the ground, you could chop them up with an axe so they're easier to move and put them aside (or give them to people with fireplaces if they're the right kind of wood). They're probably too big to compost, but I don't know too much about composting since generally more traditional composting substances are readily available in my garden.