My ex married a convicted felon. Can my ex lose legal physical and joint custody because the person is a habitual offender?

If a parent exposes a child to unnecessary risk because of his/her lifestyle choices that can certainly affect parenting time and custody orders, but 1) you need to have proof that the child(ren) are at risk, 2) the parent knew of the risk and 3) the COURT needs to be aware of the situation.

Depending on the level of the risk, Child Protective services may also become involved, but at any rate, I'd strongly advise you consult with a local domestic law attorney ASAP to insure the safety of your kids.