You're right about the difference between English Pleasure W/T and EP Open - open will judge you at the canter unless specified otherwise. Generally, open just means that any horse of any gender, size, or breed can compete under any rider as long as the correct tack is used - in this case, anything under the "English" umbrella.
Equitation classes primarily judge the rider, his/her position and ability to control the horse with finesse. Pleasure classes, in contrast, primarily judge the horse, his/her way of going and manners.
As far as how to prepare for halter classes, I do clip the muzzle whiskers (I don't like to, and let them grow out otherwise, but most judges will dock points for long whiskers in the ring, IME) but not the long whiskers around the eyes. For the ears, I gently fold the ear closed and clip away the fuzzies that stick out. That cleans things up nicely, but saves some fly protection and functionality.
For her feet, use whatever color polish you want - clear flatters some horses more than black, and vice versa. I personally prefer clear on most horses. The thing to keep in mind with your paint mare is that a shiny black hoof on a white leg will probably create the illusion that that foot is very different from the others - not something that you want in halter. Experiment a bit and find out what looks most pleasing to your eye.
I wouldn't bang her tail if it's just medium length unless it's really scraggly looking around the end. Again, it's a judgement call - whatever flatters her best. For example, if her hocks aren't her best conformational feature, banging her tail at that length might draw attention to a less-than-perfect feature. Halter is all about maximizing the positive and minimizing the negative.
If the vet shaved some hair away, there isn't much you can do about it. Most judges are understanding about little things like that - horses will injure themselves and leave marks, it's just a fact of life. I wouldn't worry about it affecting her placing in halter.
I wouldn't bathe her if it's cold out. A thorough brushing will likely be fine. If she does have staining on her white markings, you can try Cowboy Magic Green Spot Remover. It's pricey, but it does help take out the stains without soap and water. If you like, you can finish her off with something like ShowSheen, but be careful about spraying that where the saddle goes - it's slick stuff.
Have fun at your show, and do post pics if you can!!