Yes it canAny GPU can support any CPU and vice versa.
The power consumption for the GPU is 15W and the CPU is 150/125W. I would suggest a 500W Plus Power Supply.
Perhaps a brand such as Corsair or Cooler Master
The 8350 is an 8 core CPU clocked at a stock speed of 4ghz so it certainly will support a GTX 770 and it will support two also. If your plans are for the 8350 and two GTX 770's then I would get an 850w PSU. You may be able to use a 750w PSU but why cut it too close if you don't have to, the extra watts will be there if needed and will not be used if not needed. Plus if you do some overclocking you'll have the extra watts for that as well.