The answer is determined by what you mean by "joint custody" -- Illinois law defines joint custody as the ability of the parents to cooperate and communicate on issues of child raising. It does not mean that time will be split evenly or that child support will not have to be paid. The Fourth District Appellate court has evben said that if the two parties cannot agree, the court has no authority to order joint custody. The legal problems you have had in the past may effect your case -- but unless it is for child abuse or you are a registered sex offender it is unlikely the court will consider something that old.
It does not sound like joint custody will legally work, but that does not mean that you cannot ask and get extended time with your daughter during the summer. Custody does not mean that you cannot get extended and expansive time during the summer. Actually that is the ideal time for a parent who resides in another state to get visitation time since transportation back and forth during the school year will be difficult.
Consult a family law attorney where the paternity/divorce was granted. You will need to bring a motion in that court.