It depends and gets hairy here.  In some cases, as long as the judge that sentenced you didn't specify that you cannot be in civilian militias, you're okay.  In some states, this is assumed.  Likewise, if you have received a Dishonorable Discharge you are unable to join civilian militias. Always check with a lawyer before

No, you are protected by the binding precedent of the US Supreme Court case, District of Columbia vs Heller.