Well, If I were you, I'd probably go with the 6300 as it easily overclocks to 4+Ghz, I'm not sure what kind of cooling it would require, but if you find a decent cooling solution, you'd be able to overclock the CPU untill it hurts. Pun intended
You won't be able to use the FX 6300 in an mitx or matx configuration as I have not seen any AM3+ socket boards at that form factor. You'd have to go an Athlon or APU if going for the AMD option. I'm not sure about the performance of the i5 4570.
M-ATX boards for AM3+ are shit, especially if you're going to be overclocking. If the difference between the 4570 and 6300 build is a mere 30 pounds, I would go Intel.
Probably, but as one said, the Matx for AM3+ are pretty bad, so I would go Intel. If you decide to go AMD, the stock cooler will do a good enough job unless you are OCing. But if you're not OCing you should stick with the Intel.