After spinning a character to invoke a combat phase-based Advantage, does the Advantage stay in play for the rest of the game to be used again by another and/or same unspun character? I can see hording a lot of Advantages by just putting them into play and still have a max hand of 5. In other words, is there a stated limit on how many Advantages (other than the Organization) a player can have in play?

Only advantage cards that stay in play are those where it's written (protagonist, dragon seals, magic item, organizations, factions...).