I own a company with an average of 10 employees, build about 20-25 homes a year, and usually have 3-4 remodel projects going at any time. So not a heavy hitter in the industry, but not super small either. My perspective comes from that, and growing from a much a smaller company.
Most small residential companies, like mine, don't do a background check. We ask in the interview if you're a felon, and I don't think I've ever been lied to. And of course, some guys straight up say they are. We don't hire them though. A lot of our work is word-of-mouth and I don't want it getting out that we have felons in your home. People also seem to think I'm a charity and are constantly referring their friends and relatives who are "trying to get their life together". I'm not opposed to getting a second chance...it just won't be with me.
Drug tests too. We don't test, but we always ask if you can pass a test. Nobody ever said "No," but you can tell by some people's reaction that they can't. Most hard drug users may be able to fool you for a while, but they usually find a way to get themselves fired sooner rather than later. Usually by not showing up.
The construction industry certainly has its share of shitbags. You can get out of the pen and hired onto a roofing or drywall crew that day. They're generally not the ones inside your home working unsupervised though.