Someone claimed my child on their taxes and I am not sure who. How long once I refile and investigation begins will it take for me to get refund as long as I am still eligible?

If you think that you are the one who can claim the dependent, then file your tax return and claim the dependent. It you are unable to efile, then send paper return. IRS with then contact both of your and figure out who must amend the tax return and pay the tax due with interest and penalty.

Only custodial parent can claim the child. It does not count who is paying child support. Read: Your U.S. Tax Return: Child of separated or divorced parents (http://taxipay.blogspot.com/2008/03/child-of-separated-or-divorced-parents.html)

Only custodial parent can claim the child. It does not count who is paying child support.

The non-custodial parent can claim the child if either:
(a) the non-custodial parent attaches form 8332 to his return - this form must be be agreed to and signed by the custodial parent, or
(b) the non-custodial parent attaches certain pages from a court decree stating that he has the right to claim the child.

See IRS Pub 501, page 13, "Divorce decree or separation agrement made after 1984":
http://www.irs.gov/pub/irs-pdf/p501.pdf