Triggers with Lionfishes NO, imho. They will harass the lions and bite their spines.
I do not know what a Cowfish is.
Triggers with triggers depends on a lot of factors. Space is foremost, imho, but species also.
Assuredly you can mix the Menthos Family ( but those, reef-safe triggers are not my idea of a trigger )
I have triggers together ( clown triggers with rhinecanthus ), ( clowns with nigers ), ( oceanic or grays with rhinecanthus )...
Have had no luck comming Queens and Fuscus
Have had no luck comming Undulatus....i hear that if you are lucky to get a formed couple it is a beautyfull display.
Anyway, Triggers is the way to go, in salt, if you like interaction and glassbanging fish that really know you.
( i say the same about Puffers, in any case )