Is it legal for a company that is interested in hiring you, require you to submit a copy of your current pay stub?

Its not illegal.

It is not illegal. It will prove you are not lying about working there, ( since they won't do the reference check prior to hiring esp if you said not to) It also shows your current level of pay and puts them into a better position to offer you pay near what you were making.
Also shows you did not lie on application

You are not required to give it, but most likely will not get the job without it. It does not show anything they can not get with a phone call to check reference