I'll let others with more experience take the nuts and bolts questions & will tell you something you might not have realized: the army unit itself counts as an extra unit when transporting aboard ship. In other words, a 3-unit army counts as 4 for transport--which means at certain points of the game, you can be stuck trying to move an army around if you haven't planned for this. If you have only got galleys, for example, either just transport the leader counter before forming an army, or only put one unit in it & add the others when you get to where you want to go. (Of course, then you won't be able to get it back until better transport becomes available.)
As for promotions within an army from vet to elite (& I presume from regular to vet, assuming you get stuck for that) they work about as they do normally: fight a battle, get a promotion to one unit. There's a wrinkle in that you would have to win several battles to promote the "lower" units in the stack (I think) & you can only fight a limited amount in a turn. I've asked if anyone has ever gotten 3 promotions in an army & I believe no one answered yes. One is common, and 2 fairly frequent if you keep using the army turn after turn, but it looks like you have to start with elites to get more than 2 in an army. I would consider that wasteful, as armies never generate leaders on their own, so you would be giving up chances to leader fish, but if you had plenty of elites or armies & needed 20 hp, it would be viable. Elite* is a different story; they can't generate leaders while a *, so adding them to the army is perfectly fine.
kk