If I remove my spam captcha from my contact form on my site, I get lots of spam. Are each of these spam "bots" that are sending me spam counted as a visitor to my site?
Yes, they would be counted by analytics programs if they aren't filtered out via useragent, assuming they aren't spoofing them.
It depends on the analytics program. Some analytics programs read your site logs while Google Analytics relies on JavaScript code in the page. So if you're running Google Analytics and the spam bot doesn't execute JavaScript (which most don't -- they usually do a get or a post) it won't be counted in your stats.