I believe that my ex is contemplating taking me back to court to get custody of our child. I am not sure if these are taken into account when determining custody, but want to throw them out there.

1- Pays late by 2 - 3 weeks consistently.
2- Claims to not have a job or self-employed. Does he have to prove some sort of income to be considered a more fit parent then I?

Since I was given custody of the child during the divorce proceedings how easy would it be for the court to determine that my ex is more fit? Do they take child support and consistent work into account?

Any and all help would be greatl appreciated.

I'm thinking that if what you are saying is true, barring your personal situation, he is trying to scare you. It looks to me like he is going to be wasting his time and money. Then you can request that he be responsible for your legal expenses since he took you to court.
Where do you live and what is your situation, remarried, dating, employment?

He's going to have to prove why you are unfit for the custody. He is going to have to prove why the custody should be his. Sounds like he's going to have a very hard time convincing a judge why he would be the better parent as he will have to prove a better home life, stable work history, able to spend more time with the children, etc, etc. I really doubt he's serious about this as he does not know how much it costs in attorney's fees to accomplish a change in custody. You would have to be a really awful mother who lives in her car, unemployed, and a habitual drug addict or drunk for him to begin to get custody. He's also going to have to explain to the judge why he can't pay his child support on time. I am sure the judge would be interested in hearing his side.

It's probably scare tactics to get you off his back for paying late.