My ex says my girlfriend/boyfriend isn't allowed to be around my children. Can he/she do this?
Probably not. Unless your parenting plan specifically states that a certain person can't be around the children, then they can. Your ex cannot 'make up' (or enforce) rules or restrictions that aren't in the parenting plan.
My ex is exposing our children to people I don't approve of. Can I stop him/her from doing this?
Generally, no. Unless you can show that these people are harming the children in some substantive way, you cannot dictate who your ex associates with. You would need to get a court order to prohibit these people from being around the children, and courts are usually reluctant to get involved in these kinds of matters.
My ex's boyfriend/girlfriend is sleeping over when my kids are there. Can I stop this?
Probably not, unless you have a "no overnight friends" clause in your parenting plan. If you consider this to be a serious problem, you may want to consider going back to court and having a provision like this added to your parenting plan.
If the non-custodial parent refuses to return the child after a visit, do I have the right to go and retrieve the child?
Yes, normally you would have the right to go and retrieve the child if the non-custodial parent refuses to return him or her.
Contact the local police and have them assist you when you go to get the child, and have your divorce decree available to show the police that you're the parent with legal custody (and with whom the child normally resides).
If it is court ordered that you have to pick up and bring back the child, and you can't afford to do so, can the custodial parent deny you visitation?
If you can't fulfill the transportation requirements, the custodial parent isn't "denying" you visitation. They may or may not have a legal obligation to rectify the transportation issue. They should do so to promote your the relationship with the children, but chances are they are not legally required to make up for your inability to provide the court-ordered transportation.