1) Inflation theory itself occurs around t=1E-35 sec. so what I read is that it separates any two regions (points) so quickly that they don't have the ability to establish equilibrium (not that they would). If it were two hot rocks, then as you say, being close together would help even the temperature. But quantum fluctuations, apparently, are just the opposite. Putting a guy and a gal on Halmark closer together is how every show gets things right. But imagine two 2-year olds in a tantrum state; the farther apart the better. [Cat: Analogy rating...??? (1 to 10)[I recall two mechanisms for the benefits of inflation.
1) Starting out smaller allows information to be moved brfore it gets too big.
2) Expanding more allows bumps to even out more.
2) Yes. If photons can stretch why not bumps, too, I guess.