The offence, for both, is soliciting: offering or looking for commercial sex. If it's done from a vehicle, or one you've just got out of, it's kerb-crawling which is separately criminalised.
Yep. Provided you don't control it - decide who works when, that sort of thing - you can work in or visit a brothel.
Thanks to a bit of 1920s classism, no money need be involved which means that lots of places are 'brothels'. When the penalties for running one were increased, owners of places like gay saunas (where men go to be sexual with each other) went 'eek' and started doing some lobbying. So there are now two offences: owning/running a brothel (almost never prosecuted), and owning/running a brothel involving prostitution (depends on where you are in the country - some police forces will invite you onto a committee to discuss how to run them better, others will prosecute anything they find).