All in all, evolving 4* into 5* and leveling them to 70 will give you "decently strong" team that will tide you over till you obtain some natural 5*.
From what I gathered, Luck lets you obtain additional goodies from the results screen. Remember the screen with portraits of your party leader, second girl and helper at the end of the map or guild war raid? There.
One luck equals 1% chance, each of three portrayed girls rolls separately for up to three additional rewards.
People suggest that this "luck roll" is the best way to, for example, farm darumas from dojo maps: grind 1 stamina one and reap three darumas every time)
Maximum luck depends on a rarity of a unit, with coveted 99 being exclusive domain of 6* (read evolved / godified 5*)
Well, drop rates do depend on difficulty (and so, stamina cost) for dailies, so it might require quite a number of runs to see the youkai drop from lower ones. on the other hand, you might encounter a yatagarasu and snipe yourself an imperial mirror there)
And main distinction of youkai versus senki is that youkai (or "inhuman clan" as game denotes them) are not allowed in Total War. Well, that and the fact that girls need to devour them as a part godification ritual)