If a father does not pay child support and the child does not live with him at all during the year, is he allowed to claim him as a dependent on his taxes?

Is there a child custody agreement and if so, does it state he is allowed to claim him ?

Next why are you allowing him not to pay child support, turn him over to child support collections in your state if there is one, or take him back to court. Sitting in jail a few weeks or losing his drivers licence, or better yet having his pay garnished will often make them such a better parent.

Also, if the state coordinates with the IRS, they will confiscate whatever tax refund he has coming until the back child support is paid off.

Does wife need to have a medical examination to be done if going on dependent visa ?

This is NOT a tax question, but rather a legal question best posted on the LAW forum under Immigration Law.

if a father does not pay child support and the child does not live with him at all during the year, is he allowed to claim him as a dependant on his taxes?

Only custodial parent can claim the child. It does not count if he is paying child support or not.
Also, non-custodial parent can claim the child only if the custodial parent signs a release Form 8863 or there is a court order meeting the IRS requirements.