What type of flooring, site finish or pre finished, and are the floors being installed new? If using prefinished, the floors should be the last thing you do except for baseboard on one end of the room and appliances. For site finish floors, we always paint first (taking care to cover the floor) then final sand and clear coat floors after. We do the first two coats, then install appliances, then final coat for the floors. You want to make sure that your floors are in perfect condition and have as little wear and tear on them as possible when finished with the project.

Agreeing with elklaker. If your cabinets are above the countertop, then paint first from countertop up to the ceiling then install the cabinets afterwards.

Two school of thoughts, painting the walls before or after the cabinets are installed both have the same results.

We are planning on vinyl plank floors through kitchen, dining and hallway so that transitions are kept to a minimum, but will be installing an island after, since budget is way over right now - but I'm assuming that the island will have to have legs - is that correct?

That depends on the depth of the overhang. We commonly go 10" without legs, much more than this and you will need to add either legs or corbels for support.