1. Shiny wooden floor is not a problem when it comes to slipping. Your hooves have plenty of grip. Normal hooves and tyre socks may both potentially leave a mark on that kind of floor though. Poweriser (i think) did used to make non-marking hoof rubber, but I'm unsure if it's still available/in production. Some people I think have also put socks over their hooves and still been fine.
Also I should say that, if this gets to a point where it's working, and you have a regular gym, you may never get rid of me.
2. There are plenty around, some in the forum's gallery, others around google. I'm sure you can find a few.
3. Without seeing you bock, it's hard to tell. If you come down to 'boro (or I come up to NC, though this isn't likely until I have a lot more money) I could definitely help more with this. In the meantime though, it may help you to start taking smaller, faster steps as you slow down (at least this is how i slow down from running). Bear in mind that you can very rarely stop completely (without some form of support). In fact I've only ever seen it done, when people don't realise they're doing it. Instead, to stop unsupported, you must always keep your feet moving.
Hope I've helped.