Okay, I'll be a bit more expressive than I was on the other board. Sorry about that.
Yes, you can and should fire him. You have no legal obligation to provide any warnings, written or otherwise. You can quite safely fire him with no legal issues whatsoever.
None of the reasons he has provided are protected in any way. The one that might come closest is sick kid, but he hasn't been with you long enough to have any protected medical leave even if the sickness was protected, which it isn't. While there are states that provide a certain amount of leave for school issues, GA is not one of them.
There is no state where written warnings are required before you can fire someone.
Go right ahead. In the first week of employment, an employee should be on his best behavior. If this is his best behavior, you're well rid of him.

