Can you claim your child on child support?

In most states yes. However, the child will not claim it, rather the custodial parent.

No, child support is not tax deductible for the payer, nor taxable for the recipient. See IRS Publication 504 for details:http://www.irs.gov/publications/p504/ar02.html#en_US_2010_publink1000175957

I suggest that you contact your State's child support agency. When you get an interview with them, bring all the papers relating to your child support: birth certificates, acknowledgments of...

No, it is neither income nor tax deductible. see link below.

I wouldn't think so, if your child is under 18.