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The FX 6300 will game better than your previous CPU. It's by far the cheaper upgrade over the i3 which is skylake so new motherboard and DDR4 ram On a tight budget you're not going to be maxing games out any way. If need be on games that don't utilize the extra cores you can disable them and speed up the CPU a bit since there'll be less heat. I currently game on a FX9370 without issue (never tried fallout) and was using an FX 6350 for a long time prior to that.
You should look for an "E" version of FX Processor if minding TDP, PSU, and the word "upgrade" all at the same time.
Perhaps the FX 8370E if it's available in your area. Think there also FX 8320E as well.
It will game way better than that Athlon even with the current video card.
Nope, up to 32GB:4*8GB...
http://www.gigabyte.com/products/product-page.aspx?pid=4397#sp
There is 8GB stick listed on the last page.
Don't worry, 8GB will work. Why do you need 8GB sticks BTW?
2x4GB is enough.