The only LGA775 CPU's that can come close to similar gaming performance to a Athlon II X4 860K is a Q9650, QX9775, QX9770 or QX9650 @ $55 and up and being highly outdated they are just not worth it unless you can overclock. Tom's itself recommends a 860K for budget builds over Intel Dual Cores now with games that demand more than 2 cores.
http://www.tomshardware.com/reviews/best-cpus,3986.html
1.Staying on LGA 775 wont gain you all that much, 2.8 Dual core vs Quad 3ghz isn't a huge leap unless you can overclock. A Q9650 starts around $55 and up.
2. Moving to FM2+ is a upgrade but with 1600mhz RAM your losing about 25% performance, you will need to overclock to overcome this. Athlon X4 880K + Mainboard will run around $160.
3. A G4400 with a Overclockable motherboard is actually pretty good, it will have some issues with Ultra high settings and you wont be able to read Core temps but with a decent cooler you should be fine. You will also have great upgrade options. Chip + Board will run $140 or $180 to have O.C options.
4. E5-2670 "Same as I7 5960X without Overclocking" $60 will crush all of these CPU's, Motherboards are a bit pricey and upgrades are expensive. Chip + Board $180-$220