How important are Big League Managers? And what about 1st base & 3rd base Coaches?

You are the manager as you set the lineups and rotations. In the help menu: "1B coaches and 3B coaches impact hitting and baserunning ratings." As with anything, the coaches with better ratings in those categories will help, while bad ratings will hurt your team. Bench coaches have a smaller impact in the majors (taking over when a manager is ejected), but seem to have a larger impact in the minors and prospect development.

The answer in my experience is "not very", as long as they aren't bottom of the barrel you're fine. I haven't noticed great coaches (90+ ratings) providing any meaningful difference. BUT, bad coaches (less than 60 or 70 at ML level) will def mess up your players' ratings.

Don't overpay (exception being FI, always pay top dollar for a good FI). If you don't need a FI, you can get away with $6 or $7M for coaches in most leagues if you wait until the very last coach hiring window and offer half of their asking price. If you do it this way you get very solid coaching options for min salary or close to.

Only time to overpay for a non-FI coach is if you want to offer a max contract and know you'll need the coaches to break the tie breaker (PC and BU coaches for pitchers and HC and FI for position players)