Kirk likely wouldn't have put any other member of his crew in that position because he doesn't trust them like he trusts Spock. Spock leaving Kirk behind says nothing about how he feels about him, logic would dictate he do his duty and follow the Prime Directive.
This Spock is a decade younger than the Spock we see in TOS, I think it would be a reach to call him the best first officer in the Fleet just yet. Enterprise is likely his first posting as an XO based on his age.
Trust doesn't have to mean they're friends, and Kirk had to ask McCoy what Spock would do (a "rhetorical" question for the benefit of the audience, maybe you would suggest
). Re Spock as first officer, I did say "potentially at least". To a degree I was kidding of course, but Kirk may just have seen something in Spock that he thought showed promise (or better the devil you know!).