More specifically, do corporations who claim to do background checks always do them? I've heard that some only do if the applicant puts something down under their criminal history.

Most do....depends on if you're dealing with cash or how high a position you're going to take.

Yes, your background is checked out thoroughly. Employers, to protect themselves against the possibility of hiring someone with less than a stellar background, will check out your past work experience, your credit history, and any criminal background.
Thus, if you screwed up as a young free spirited person, who was out to impress, without a care in the world, then you will be very much screwed as an adult looking for honest work.

From what I've seen yes they do. It protects them from a bad investment, a thief, etc. It's worth the money to the employer.

If they say they will conduct a background check, then they will. I used to conduct background checks on all employees at my old job. You really have nothing to worry about unless you have a criminal record.