"Should you even install the hardwood under the base moulding? Aren't you going to leave that space for expansion?" Clearance depends on the type of floor being installed. Narrow strip hardwood flooring (3 inches and under) barely moves 1/16 of an inch per board. The movement occurs on the un-nailed side of every board so it dies not build up over a large span. Wider boards need some additional room, but it is still pretty small until you get up to 12+ inch boards. The length of hardwood flooring (parallel to the grain) barely moves at all (1% or less). Some engineered floors that are installed as 'floating' behave as a very large piece of wood with the only room at the edges. They can need significant gaps . Other engineered flooring has counter-plies in the layup, like plywood. These need some room (just like plywood) but it is still not very large. I would never bother installing baseboard before floors. It is just an invitation for things to go wrong, as the OP found out.

A small gap between the wood and stone is what is recommended...that gets carefully filled with an elastic caulk. There is usually some disagreement amongst installers as to how much expansion space is required for a wood floor to perform well and not to have problems due to being locked in place. I have seen solid wood floor cut net to vertical surfaces and they never had a problem. However, I can definitely think of houses where cutting net to vertical surfaces would be risky, if not downright foolish. Flexible caulk of a close match is what I'd use.

kitchenkelly (and everyone else) should I have them maybe cut that tapered trim (I don't know what it's called) maybe in half and then do it picture frame style? I sorta need to make up my mind by tomorrow morning when they come to finish things off. I like the tapered edge, not so fat as the quarterround, so maybe that's the way to go.

The baseboard and 1/4 round, with mitered corners, should continue around the corner to meet the cabinet, as everyone else has said. Is that white vertical line on each side of the doorway the edge of painted exposed sheet rock? How will that be finished?