The answer to both questions is yes, but let's talk about getting the mother's permission first.
Even though the child may qualify as your dependent. The mother (or the father) may claim them on the their tax return. If that happens the IRS will contact you and the other party to determine the actual status of the child. That may not be an issue you wish to deal with. It is best to first contact the parents to confirm that they are not claiming the boy.
Yes, if the child lived with you for the entire year except alternating weekends, that is considered the full year.
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