To be on the BHS register you need to pay an annual fee (amongst other things) so will only remain on the register if you pay this fee.
I am not sure if there is any other list that is available for the general public to view, although I'm sure BHS must keep a record for themselves of people who have qualified?
Yes you have to pay around £136 per year, this gives you liability insurance to teach.
You have to pass a criminal records bureau check, go on a first aid course and sit a refresher every two years and also a child protection evening course. How up to date it is I dont know - one would hope annually updated otherwsie there would be a load of old school instructors who passed their AI ten years ago but not bothered to keep there refreshers up to date which would make a mockery of all those that do!
I believe that you don't have to be on there if you are permanently employed rather than freelance as your insurance etc then has to be sorted by your employer.
Personally I think there should be two registers, one for freelance instructors with paid up insurance etc and another just so you can check qualifications. I want to be able to check what level of instructor I am paying for at a riding school independantly, after all why should I pay the same for a BHSPTC as for a BHSI???