Hopefully, your middle name is different. When you are extended a job offer contingent on references (like the previous answerer said, they don’t check references until they are READY to hire you), just say that there were two “Joe Smith”’s there and emphasize your middle name, hopefully that’s different. If even the middle initial is the same, emphasize your birthday. Hopefully that’s not the same as the “bad” “Joe Smith”.

Depends, did you have the same boss? If the company closed you sould leave the name and number of your previous direct supervisor and call him to make sure he is aware that you are putting him down as a reference. As long as you left on good terms this should work. During interview if asked why you left be honest and when they call to check your reference they should have a very similar story.

I would absolutely bring it up, especially if you think you are losing jobs because of it.

As a side note, most employers don’t call references until after your first interview. Some use it as a screening tool for applicants, but it’s kind of a hassle to call around and most will wait until they see if they like you in person. Also, if you’re worried about it, have a friend call your previous employer and pose as a potential employer and see what they are saying about you. I have done that in the past just to see what the word is about me…