Can I use a PS4 controller on a PS3?

It is possible. Eurogamer says you can do it wirelessly (How to use a DualShock 4 wirelessly with a PS3), but you definitely can do it wired, which is better because it reduces input lag (especially if you have a gold USB).

However, you cannot get to the XMB menu, and it seems to only work with Sony controllers (officially licensed ones don't count), as I have tried different controllers like the Scuf Vantage 2 and it won't work.

You can also use some of your PS3 controllers / peripherals on PS4, which are known as "Legacy Controllers." Hori products, particularly the Fight Stick Mini for PS4, have a setting that can switch between PS4 and PS3.

And a lot of products are either already compatible on PC, or there's drivers that will auto download, or, depending on settings, auto setup on PC (my old gold headset works on it). The DS4, even with the Official Adapter / Dongle, won't work properly without DS4 Windows / Input Mapper.

So basically, yes, it will work. And unlike some people state, it doesn't just navigate the XMB.

No you cant. Only allows you to scroll through xmb.