The Order of Child Support should state who gets to claim the children for taxes. Depending on your relative income, it might make sense for you to agree that the father can claim the child, if he agrees to pay you for the difference. That would depend on a careful analysis of your and his income.
The IRS exemption goes to whoever is named for the exemption on the child support order.
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This is a question most lawyers cannot really answer correctly. The answer is "yes", of course, because he "can" try to do anything. There are a few approaches to this common problem. First, think about the IRS -- all they care about is that 2 people dont use the same social security number or tax dependent, so often, the ruling by the IRS is "first to file gets the exemption," and then they reject the second tax return because the social security number conflicts.
Second, the courthouse is the place to go if someone violated the tax exemption language. Even if father claimed the child, and if the IRS says, "sorry Mom, he was first," then that does not mean you do not have a remedy for it. For example, I have filed contempt proceedings on this issue arguing, "Your honor, my client was deprived of $2,000 due to the loss of claiming the exemption." In the case I filed last year, the court said: (a) Dad has to pay mom $2,000, (b) Dad has to pay mom $2,000 for her attorney's fees, (c) Dad has to pay a civil penalty of $100 to the clerk of the court due to contempt, and (d) mom gets the next year as well.