The contracts increase the number of items the shops sell and make it possible to buy (I think) every shop in the game, I think they may also increase dev level (if it isn’t already maxed) and reduce the costs. There are also others that provide permanent passive buffs like increased walking speed, lure (target enemy + down on D pad) distance, reduced affinity loss, etc. You can’t ‘use’ any of them except when they’re story related (Ether cylinder powered the door in the ship in Chapter 1) and you use those on the field (ie ‘insert ether cylinder’) or other menus (overdrive protocol on ‘manage blades’). The merc mission rewards are permanent unless you wipe your file completely clean (ie ‘new game’) and stay with you in new game+. Story and quest related ones will disappear in NG+.