You probably noticed as you accrue points in the Splatfest, your title changes. Fanboy/girl->fiend->defender->champion->king/queen. Since king/queen is the last title, a lot of people stop there and don't play the splatfest anymore. When the Splatfest is over, depending on your title, you get sea snails (even if you're on the losing team, albeit fewer) that you can use to add slots to equipment without max (3) slots, or to reroll a slot for a new ability.
Food tickets, easiest way is probably playing Salmon Run and getting them via bonuses (each 100 point threshold gets you more bonuses, can be money or ability chunks or tickets, etc), though I think you can also get them via single player/the expansion.
You'll have to specify what you mean by special gear rewards. Each month there's a new piece of equipment you can get only via Salmon Run. There are also pieces you get through single player (one of the hero replicas for completing each stage with the corresponding weapon) and in the expansion by collecting sets of mem cakes.
