My suggestion would be to go and look at homes with different ceiling heights. If the ceiling is too high, while the room may look good, it may feel uncomfortable and people don't know why. Psychologically, we humans actually don't like living in places with huge ceilings. 8 foot ceilings are very low. In Australia it is not the norm, and while it reduces building costs, it can make a house very claustraphobic. 9' is a nice height. I would not choose 22' high in your family room. Personally I feel it is too high and as the family room it will be 'very lived in'. My family room ceiling is 10' (so is all the ground floor of the house) and the height is very comfortable.

Definitely go with 10-foot ceilings with 8-foot doors. 9-foot ceilings are totally useless and do not give you any sense of increased volume. I can't tell you how many homes tried to compromise with 9 feet and ended up not making the home feel more custom and spacious. If you do go with 9 feet, putting in tray ceilings will make that height feel more gracious.

A tall bed might work in the room, but it would help if you added pictures (of the room and the bed) and a floor plan. If the bed is peaked like the ceiling it might look weird, but if it's a canopy style it might look really nice.

I can see no reason for your issue. There is nothing wrong with an eight foot door on a nine foot room, header is not exactly a "feature" !. They are more expensive of course. If you WANT the mud doors at a standard height, and are concerned about a matching laundry look? Drop the ceiling in the laundry to nine feet. I can not see a reason for ten foot ceilings in a laundry. Nor can I see any issue with those eight footers, in an entire mud area. Don't wring hands on a non issue.