With good shoes and suds/broddar whatever they're called, the horse doesn't slip. :P I've cantered (read ''the horses took off with me and my friend as fast as they ossibly could'') 2 miles on a road with several inches of pure ice on it. The horses didn't slip once because they haad the right stuff on their feet :P
(I fell several times tho when leadig them back home half of the way, not wanting to get up before they had calmed down properly.)
The backside is that a too good grip can, with time, damage their knees as the natural movement (of the hoof sliding a little on the ground) is stopped too abruptly and the joints take more impact. So it's important not to overdo it, and at the same time not do too little either or they'll tense up and risk injuries.
Unless you fully clip the horse, or it's being sick, it won't have any problem with being bare when you ride. And if you clip it, there are blankets for riding :P
My horse stumbles quite a lot too (never learnt to lift his feet;) but with the right shoes it's not worse than in summer time. And it's not dangerous for them to stumble, unlike us they have 3 other legs to keep balance with. But if they feel like they're sliding and slipping on the ground, they'll tense up, and that's what's more dangerous becase then they'll stumble and have half cramped muscles at the same time.
But if you're not confident in riding, you shouldn't do it. It'll just be sensed by the horse and then you both will be nerous, so stay in situations where you can relax :) Or meet your ''fear'' but note more than a bit at a time; you could for example go trailride and stay in walk and on the roads or paths untill you're confident and learns that the horse won't stumble, or if it does, it won't get hurt and you'll learn to stay in the saddle. Going off road is great excercise but don't let a misstep scare you off :P