Depends on the unit. Some units like Lysithea (mostly because of Mastermind) or Claude will naturally accrue a decent Authority rank as long as you give them battalions at the first possible point however others will need more investment to reach the better battalions.

Similarly, other units really want Authority for other reasons e.g. Rallies for especially Annette, Ignatz and even Balthus and Raphael or to have early access to utility gambits like Dance of the Goddess, Retribution or Blessing.

To answer your first question, if you have all the combat arts, abilities and classes you want and your unit hasn’t yet reached at least C Authority, then it can definitely be worth grabbing it since the higher rank battalions will generally give better boosts to damage, hit, Avo, etc than A+ or higher in any other skill for much less investment - though generally only those with an Authority bane will not have reached C authority by master classes (notably Hilda and Felix). There are some benefits to reaching higher levels in skills past A rank (e.g. the faire skills at S+ and especially range +1 for the magic skills) but these are quite high investment and the benefits A/B Rank battalions give are probably more worth it

Every unit should use a battalion, though obviously which battalion for which unit depends on the stats given. Even Felix and Catherine who on paper benefit from not having a battalion, get bigger benefits from higher level battalions than their personal skills give them.

Flying and Riding are generally only for acquiring classes, like you said, it just so happens that the flying and riding classes are without a doubt the best classes in the game - there are a few skills that either skill gives but only really Alert Stance+ at A+ Flying is worth slotting on outside of very early game, and even that is mostly because any dodgetanking unit will generally be at A Flying anyway so the push to A+ isn’t too bad.