The moving company is only as good as the agent doing the move. I could recommend United because the United agent in our town is GREAT but I know of several United agencies that are horrible. The best thing to do is ask people in your area who they have used and would they use them again. Ask if they had any problems, etc.
As for cost, it really depends on how much stuff you have not how many bedrooms. The cost is determined by weight so you could have 10 bedrooms and only have furnishings in 4 of them, etc. It is really hard to even give you a guess because if they don't have much stuff and share a truck with someone else, it will be less expensive then if they had a whole truck to themselves.
Are they going to have the moving company pack everything and move it? I would suggest that they get FULL REPLACEMENT COST insurance. You don't want to submit claims to your homeowners policy and if you don't get full replacement cost, they depreciate your goods so you might get $15 to replace a 3 year old TV.