I will prove to you that God exists

[Sheldon]

It's pretty fatally flawed. He keeps chanting about "incredibly low probability" and "narrow range". That'd be true if he was trying to get humans by chucking darts at a board at random. We don't know what range of values is possible, if any, we don't know how likely our values are, we don't know if they're random or independent, and we don't know what happens when our numbers don't come up. Lacking that you can't do a useful evaluation of the probability space and his entire argument boils down to one of incredulity and ignorance.

It is certainly an argument from ignorance if Drew is uninformed about the possible alternatives.

His argument seems to me to be that anthropic multiverse cosmology is a reductio ad absurdum of materialist thinking, that a logical extrapolation from inflation and string theories leads to nonsense.  But that argument can't be true if there are other logical alternatives, even if you grant that the multiverse doesn't make sense to begin with -- a big "if."

As Boru said, Drew keeps jumping to unwarranted conclusions.  He can't seem to hold the unknown in his mental grasp.  I suppose that is true of a fair number of theists.  They want to know.  They think it is important to know.  So they jump at some "answer" or other.  Perhaps the alternatives involve too much work, and they are just intellectually lazy.  It's hard to say.

"He can't seem to hold the unknown in his mental grasp."

Exactly, why do theists like Drew, think an absence of knowledge is a sound basis for belief, rather than the opposite. What's worse when we and he admit we don't know, he shakes his head, rolls his eyes and says then it must a deity, and when we refuse to share his belief, he insists we must be making a contrary claim, and insist his claim has merit if we can't offer an alternative, which is the very definition of an argumentum ad ignorantiam fallacy. When his arguments are exposed as fallacious, he first ignores it, then denies it but just with handwaving, then repeats his fallacious arguments, and lies, denying that you explained why it was fallacious, then finally goes with the "you must all be idiots" argument and flounces out. 

Agnosticism is not a reasonable basis for theistic or deistic belief, rather it is a sound reason to withhold belief. As it would be in any other circumstances, facing any other claim, god claims are no different. Drew's bias was evident throughout, but never more so than when he admitted he does not submit all claims to the same standard before becoming credulous. Just why the phrase "I don't know" causes religious apologists to panic, and shriek as if they've won the Templeton prize, is anyone's guess.