Recently there has been some negative activity on my credit report that I feel is not mine.

If you feel it isn't yours and is illegal, then you would have to have to investigated, but we all get carried away with credit and what you see could be justified and you just never noticed.

As for refinancing your home, that would be up to your mortgage company on past activity and present income.

You probably can but your interest rate might be a bit high however for the last year if you have made on time payments for your bills and have a low debt to income ratio you should be fine. Meanwhile get a copy of your credit report and make sure it is correct.