If our universe is so complex that it couldn't have happened without a designer, how is it possible that God does not have a designer?
Why can't a circle have right angles, or a fish have feathers?
By definition, a circle has no right angles and fish have no feathers. Likewise,
by definition God has no designer. If He did have a designer, then His designer would necessarily be superior to Him and thus His God. All you do by positing a designer of God is set the First Cause back a step. In any case, the concept of God, at least in the Christian worldview, necessarily entails that God is causeless. God cannot be God if He is not causeless just like a circle cannot be a circle if it has a right angle.
If God is special in that he's eternal and causeless, why is it not that our universe may be eternal and causeless?
Because God is a Cause, the First Cause, in fact, while the universe is the
effect of a cause. Mainstream secular science has established by way of the Big Bang Theory that the universe began to exist a finite time ago, which points very clearly to the universe being an effect, not a cause.
"Ex nihilo, nihilo fit." Also, there are serious logical problems with holding that the universe is infinite. For example, it is impossible for the universe to have arrived at the present moment because, if the universe has an infinite existence, there is an infinite span of time existing before this moment which is, because it is
infinite, impossible to cross.
If God is beyond human comprehension, why is it not just that our universe is beyond human comprehension?
It is very likely that there are aspects of the universe which are beyond our comprehension - at least at the moment. But the universe is not utterly beyond our comprehension. In fact, there are many things about it that we do know. And some of those things indicate that the universe is finite and caused.
Selah.