Being an HR professional I would never ask for pay stubs to confirm someone's wages. What you are being paid is really non of there business. I always ensure that our Total compensation model is comparable to the industry and will work with the candidate to get them something we both can agree on.

I honestly wouldn't want to work for a company like that.

Also if your looking at even considering a counteroffer form your current employer don't!!! If they value you as an employee they should not wait for you to resign to counter with more money.