1. If you aren't sharing exp between classes, msq should be enough to get you through stormblood. Though if not, roulettes are your best bet. Also if exp is an issue now, you definitely won't get through shadowbringers without a bit of grinding (which is generally standard).

It is not mandatory to level or even unlock the chocobo for combat purposes. It's just handy to do so. I do recommend healing over damage though but its 100% up to you.