1) Is number of admins from trained citizens capped? It looks like it is, but I'm not sure. Haven't gotten there yet. Would be good to know.
Yes, you can only have 10 Administrators at the same time. However, you can promote Administrators to Ministers, they also give admin points, or Mobsters, thus freeing up space for more of them.
2) Cities still. I've finally built them, but I'm not convinced 4 food is the only requirement, nor am I convinced you always need 4 food. Maybe there's a bug? I think happiness is involved too, btw.
The cost for a City is 4 food and 10 Duranthium. Approval has no bearing on building Cities.
3) Adjacency guide? I wish each improvement listed what it would benefit from. Sometimes I have to terraform 'blind.'
In the tooltip for improvements, just under their name, it lists their type. They receive adjacency bonus based on that, usually.
4) Are old ship designs auto-carried over to Crusade? I don't think I'm seeing all of mine.
Yes, check the ship designer options for it to show over mass designs and others.
5) I'm not used to the ai beating me on wonder construction. Any wonders that are must-have on ludicrous-no-victory-conditions?
Depends on playstyle and what victory condition you are aiming for. Eyes of the Universe is good for knowing the general area of you opponents.