I would not get a skylake x CPU for gaming, but they can game. If you are not planning on overclocking, then you should go with the 7820x as it has higher base and boost clocks. Those will help you in gaming.
If you are going to overclock, then you could settle for the 7800x, but I dont know why anyone would want that CPU now that the 8700k has been released. So if you need more cores than 6, go with the 7900x and some good cooling so you can overclock.
The PSU really depends on the rest of your system and if you are going to overclock. For a 7900x and a 1080ti, overclocking both, I would go with a good 850 watt PSU.