Meter and service panel separate from house
This is a question that only your AHJ (Authority Having Jurisdiction, aka local inspector) can answer for you. There is no specific NATIONAL code violation in doing so, but there could very well be local codes prohibiting this kind of setup, having to do with life / safety issues, fire fighter access, maybe even utility requirements.

If it IS permitted, you would NOT install a new set of ground rods at the house, you would run the EGC (Electrical Grounding Conductor) from your service entrance ground connection back to any local "sub panel" that you install at the house. Likewise with your Neutral conductor, and in this case, because the main is where the Neutral is ALREADY bonded, you would NOT bond the neutral again at the house.