An FX 8350 would probably be able to keep up with a 380 and it is the same socket motherboard. Keep in mind that some AM3+ motherboards weren't built to handle the extra heat that comes from an 8 core FX model and will end up throttling. Whether you would be affected by that or not depends on what model motherboard you have.
Not as much difference as an upgrade but it can't hurt to try since it's free performance. This is assuming that you have decent cooling though. It isn't typically recommend to overclock on the stock cooler.
You don't need to change your CPU. A 4300 is close to balanced with an r9 380 as is. You may not be quite pushing the card to full capacity in CPU-intensive games, but you can get there with a mild-moderate overclock (a $30 investment in an aftermarket cooler)
An FX 8350 would probably be able to keep up with a 380 and it is the same socket motherboard. Keep in mind that some AM3+ motherboards weren't built to handle the extra heat that comes from an 8 core FX model and will end up throttling. Whether you would be affected by that or not depends on what model motherboard you have.