Actually, Kaidan is Shepard's superior officer now.

She is still a Lt. Commander. He's now a Staff Commander (1 rank below Captain.) I'd have to look at the codex entry on ranks in ME1 again but there are 2 Commanders ranks and I'm fairly certain that's how they're listed.
As for the Council and Human Spectres, if you do a playthrough as Paragon and then regain Spectre status, it makes little sense to add in another human Spectre. Their conversation with you, and reasonings behind reinstating your status actually means that they do indeed support you. They simply cannot support Cerberus. As far as the public is concerned though, the Council is behind you personally.
I could see Udina pushing for a new human Spectre, but I'm not sure about Anderson or the Council. Of course they could mix things up a bit with new information, but given what we have to work with now, it's really iffy on how it'd fit in.
Lastly, it really depends on if Kaidan wants to become a Spectre or not. His allegience is to the Alliance, first and foremost. The Council never really instilled a sense of loyalty into him like it did for Shepard. He wasn't as distrustful over them as Ashley was since he did try to see things from their perspective. But the incidents after Virmire and the events leaidng up to the final battle and then later at the intro of ME2 really really shook his confidence in the Council.
I mean afterall, what would Kaidan say?
Kaidan: Shepard, the Council and the Reaper threat. They don't take it seriously. I just... don't know...