Going to Pompeii from Rome is at least three hours each way by car. It's a very long day, plus the heat, so I can't imagine doing the Archaeology Museum the same day. How about an overnight in the area?
After the train from Rome to Naples, you have the chaotic Circumvesuviana train to Pompeii. We rode this last week, and it's packed like a NYC subway, no air conditioning, probably standing room only, and the "schedule" is more like a suggestion. In addition, the train can have delays en route, so that it takes even longer than anticipated.
Private transportation is an option, but I was amazed by the congestion in the Naples/Pompeii area. The roads are small and the demand for the roads great. You spend a lot of time sitting in traffic. There is no efficient, fast way to travel thru the area.
Mondo tours offer shared tours at 11:00am. With the size of your group, you'd probably need two guides, as their groups are usually 6-10 people. RS recommends Mondo in the guide book. Mondo also has shared transportation; look on their website to see if they could meet your needs getting to Pompeii.
I didn't answer all your questions, but attempted to explain some of the basics in the area. It's hard to imagine, until you're been there.
Pompeii is amazing! Be prepared for uneven " sidewalks", wear sturdy shoes, and food and refreshments are only in limited areas.
Have a great trip.