They'll be pretty close in most situations, but anything with more than dual thread support, the 6300 is going to walk away from the i3. I have a 6300 in one of my machines and it's a great CPU.
Either CPU will run those games as well as the GPU lets them, but they'll vary otherwise in that the i3 might be a bit faster in normal tasks (browsing, word processing) but not noticeably so. High powered tasks go to the extra cores on the 6300...so the AMD would be my suggestion.
The FX 6300 is great processor, and it is almost at the same price as the Intel's i3. I would definitely get the FX 6300, those six cores might come in handy with future titles. From all the games that you have listed, Skyrim prefers Intel processors, but there is no point to buy i3 just for Skyrim.
About the motherboard, get the Asus M5A97 R.2.0 or the Gigabyte GA-970A-UD3P, both great board for the FX 6300 in the range of $80