Yes, but not in the way that you think. When you say "has a reason" that means it has a cause. Causality is only applicable if existence is presumed. Ie. if something exists (the universe) it can give cause to something else. Otherwise it cannot. And since existence is only possible inside the universe (there is no such thing as "outside"), nothing can logically give cause to the universe. Why? Because "the universe" means "everything that exists".
Besides, even if you go along with it, god must also have a "reason" to exist, otherwise you contradict yourself. If you were to say instead "everything natural has a reason for its existence, but not everything supernatural" then I would obviously have to ask how do you know this.
The universe is not necassarily all that exists. You merely assume this.
Now consider three alternative cases that could possibly explain the fine tuning of
the universe.
1) Natural Law.
2) Accidental.
3) Intelligent Design.
Consider that the universe is in perfect balance. This is an undisputed point. Even Hawking himself said, that the slightest change in the velocity of the Big Bang would cause the universe to collapse into itself resulting in a big ball of fire.
First, the universe does not have to be the way it is. It could have been otherwise. Thus The first alternative, Natural Law, is not very plausible.
Now the second alternative, chance. The problem is that the odds against the fine tuning occuring by accident are so great that they cannot be reasonably fixed.
When you compare the range of possible values which the fundamental quantities permitted by the Laws of Nature could have taken with the range of life permitting values, you find that the range of life permitting values is incredibly small in comparison with the wider range of assumable values.
The probability that all the quantities would fall by chance alone into the life permitting range is vanishingly small. We now know that life prohibiting universes are vastly more probable than life permitting universes like ours.
Third alternative, intelligent design.
Premise 1. The fine tuning is either due to law chance or design.
Premise 2. It is not due to law or chance, therefore it is due to design.
I don't understand this. Is he saying that something which begins to exist cannot be its own cause for existence?
Yes.
Impossible. Causality is only possible within the universe.
That is only your assumption. That does not make it valid.
"personal creator" as in the Christian god? Absolutely not. Even if we were to agree that there is something which created the universe (a burst of energy, or something similarly strange), there is no reason to think it's a consciousness.
Because of the fine tuning mentioned above and the Moral Law argument I proclaimed today

We reasonably assume that this Creator was a consciousness.
No. Even if "someone" (let's say for arguments sake that it's a consciousness) created the universe, that gives no reason to think he still exists. He could have died.
A creator that is supernatural is limitless, He cannot die because He is unnatural.
No. None of these attributes follow logically from "something" that created the universe. And certainly not uncaused, like mentioned already.
See above.
Time has no meaning outside the universe, the universe is what defines time. However, I admit I find this one argument of mine less persuasive than the others.
Why is it not the other way around. I think time is what defines the universe. Don't you think that my assumption makes more sense, can the universe exist without time?