I can only talk about one agency that I know the ins and outs of, and it is in Edinburgh - the ones in London may be different.
At this agency, yes you really have to go to whoever they send you to. You wouldn't be able to refuse because he's Asian. When a guy books they take his name and address or name and hotel room number. And a mobile number - they don't ask for landlines. No, this agency doesn't vet clients....they say they do, but they don't and the amount of jobs that are wind ups or don't answer the door, or just want a look are massive. They do have a driver, so you do have someone with you sometimes, but not always. You are also (and I think this is the case with most agencies) 'on call', so if you say you're working Saturday, that means if she phones with a job for you on Sunday morning at 3am, you will be expected to be ready and with the guy within half an hour, so really you either have to sleep with your make up on and just doze in the chair, or stay up. You can't really go to bed. If you miss a job because you're sleeping, you won't get any more jobs for a while in retaliation, that usually lasts 7-10 days, but if someone specifically wants you she might lift the ban quicker. This particular agency also checks viva st and adultwork looking for girls who work for her.
Funny when you think that they agency actually works for you, isn't it?
This is just one agency though, but I can only speak about the one that I know. I'm sure there are good ones out there.