On a budget PC, will the CPU and GPU fans be enough to keep the whole system cool?

Absolutely. Each of these components come with a cooling system already installed. It does go without saying that any PC manufacturer would not let out any of their PCs if they cannot simply cool down at all. that would be disastrous. So yes, even a budget PC has a cooling system in place, even if you assemble it yourself. You may replace or add up yours if you need more efficient cooling. However what's already in place is sufficient enough.

Depends on how the case is built and how the CPU fan is oriented, but generally you want at least one exhaust fan. Most cases come with at least one, but if not they're cheap (less than $10 a piece, generally).

Possibly, depends on too many things.

Buy it, run it, check the temperatures and see.

You won't burn it down that easily.

Depends on the case design and where the case is located.

Probably - I really hope your PSU has fans as well, otherwise you'll burn your house down.