if you're a new graduate they sometimes do--if it was years ago, they usually don't. If it's an important/high-level job or if they've been burned before, they'll check no matter what. People are always being caught out 20+ years later, after they've risen in the ranks.

Be honest about this kind of stuff; employers more often than not check this out.

I had to submit a copy of my diploma and transcript to my current job, perhaps because I am a recent graduate. However, I got the impression that documentation is required on all degrees/professional honors/etc for all employees. However, my company is a little odd, so YMMV.

Well, grad school ain't gonna happen until you finish, so if you go the lying route, be certain you aren't going to want an MBA or an MLS or a PhD anytime ever.
Other than that, I have a friend who didn't quiiiiite graduate (due to a not-the-right-credits technicality; she went to college for 4 years, took enough credits to graduate, but didn't have one required class for her major). She's in her very early 30s now and doing just fine, thank you, in a field in which you really are supposed to have a BA. Neither of her 2 employers in said field (very large companies indeed, I might add) bothered to check.

I'd not lie about it. These things can bite you in the ass. Tell 'em when you expect to get your degree.

I personally request official transcripts from the university in question.

So yes ( at least in my case), it would be a bad idea to call yourself a graduate as I look at your unfinished transcript.