I always paint first. No worries about dripping on a new floor.

However, I wonder about the baseboard in such a case. Will more of the baseboard be exposed with vinyl? Will the floor installers be undercutting the baseboard to insert the vinyl?

Make sure to tell the painter to paint the baseboards very low, pushing back the carpet as much as possible.

moving furniture is always going to be last and always runs the risk of scuffing paint.

You may have to touch up when everything is finally done. Make sure the painters leave the leftover paint.