Wikipedia on warts
Yes, they are contagious. They are caused by a virus. However, as contagious illnesses go, I think they are not terribly contagious, otherwise everybody would have lots of warts. I had one on my foot for a couple of years; during that time I never developed any other warts on my body, and my wife never developed any either, despite using the same shower and sharing plenty of body contact.
As far as getting rid of them:
I had my wart frozen a couple of times at the doc’s office, but it refused to die. Third time was to a different doc. He used a disposable blade to shave the area down really, really thin (he stopped when he finally drew a smidge of blood), then suggested Compound-W. I applied it every evening after a shower, then covered with a band-aid, leaving it on until the next night’s shower. Did this for a week, and finally this past Saturday I was able to peel off a thick patch of skin that included the wart. There was new skin underneath, so no blood, no pain.
I never had much success with Compound W when I was a kid, but then I probably wasn’t doing it right. I’ll have to see what happens to that area over the next few months, but for the time being, it sure looks like it’s gone.