Pick your counter top first, then your backsplash, find something your fall in love with, then pull your cabinet colour out of that. Your linear areas should complement each other (ie flooring and countertops) backsplashes should be the accent but complimentary and cabinet colour pulls it all together.
As far as installation, we prefer to complete the onsite painting prior your countertop going in because many installers use non-paintable silicone to seal the top to the cabinetry, this makes it difficult for us to paint. The order tends to be, but no hard and fast rules, flooring, in home painting, countertop, backsplash, finished doors and drawers installed, wall/room painting.
If your countertops are already installed, we do have tricks up our sleeves to combat this, so it’s not an insurmountable issue, this is simply our preference.