You really aren't an expert in background checks are you?!
I think you mean checking references. It's a whole different kettle of fish.
Since you still haven't told us which firm it is, we still can't help you. Some do, some don't. Many do sporadic reference checks.
So, to reference your replies:
2. They don't do a REFERENCE check if you don't have any references to check. In your case it sounds like your application does have details on it
3. Not unless we have specifics and / or the sufficient motivation
4. Silly child, but you probably will after the first 6 months of your career sweating on that call / email asking to meet someone from HR with your line manager present