Can I still have a handgun on me and in the house if my wife has misdemeanor and is on probation?

This depends on the conditions of her probation the answer is maybe.

If the court did not take away her firearms rights, she may have a handgun.

Your wife's problems are generally occurs, and do not affect your rights. However, if she is on a misdemeanor probation I would have to look at the terms of that probation and order to determine whether or not your exercising your rights are potentially putting her at risk. No easy answer and no from opinion without all of the facts and circumstances.