Agencies usually charge 30%. The one I worked for briefly was less (about 13% when I worked it out!)
If you use a driver (IMO essential for home visits) the agency may send him but you'll have to pay his fee separately, so find out what he charges.
The client will pay you the full fee (in cash, no cheques), then you give the agency their percentage. Some agencies allow clients to pay by credit card and send you the money later, but don't accept that kind of arrangement until you've done a few jobs and feel that you can trust the agency, as a lot of scams work that way.
As for the hair, maybe you could make it look different enough with a creative do or a hat? Or only do headless shots, that's not uncommon so shouldn't be a problem.