Is a mother considered unfit if she lives with a man (not the father of the kids) but doesn't marry him?

Have you never heard of co-habitation ?

This would NOT be considered as unfit.

No, this would be considered normal everyday life for a large group of unmarried people

One is considered an unfit mother when she mistreats her child, or allows them to be mistreated by others. If this is not the case, then you are not unfit. Since you ask this question, consider if you may be too caught up with what others (ie family, friends, or society in general) think... or that the other person you live with may be unfit to parent. If the latter is true, then you should check yourself. If not? Then you should stop worrying about what others think - the only person that matters when it comes to parenting, is the one that you are parenting.

Cheers

So long as your living in the U.S. or any other modern country its not. Some of the more third world countries it might be illegal. Or any of the Muslim nations.