I agree that from a purely rational standpoint God is not "necessary" for the system to work as we understand it, that's why I get frustrated with Christians trying for proofs.
Knowing the logical arguments against, I find them unconvincing for the same reason He is not falsifiable scientifically; there's no reason to presume God's reasoning is completely comprehensible to the human intellect, in fact within doctrine it is made clear it is not.
Not sure how belief introduces complexity though, the few questions it raises are more or less irrelevent to those who don't accept His existence in the first place. For instance, the "problem of evil" is only a problem if you believe in or are trying to disprove God (looking for trouble intellectually so to speak). If you don't believe in or don't care about God, you should just go with naturalistic explanations which are perfectly sufficient. No complexity QED.
Anyway, you've NEVER heard some idiot, usually an aggressive atheist pot stirrer, claim that science makes God impossible or some such? You need to get out more. 
Complexity: One of the traditional arguments for God is "a "Prime Mover" is needed, or a first cause. This is because every action has a cause, and that causal chain leads back to infinity. Since there cannot be a causal chain leading back to infinity there must have been something outside of that causal chain and thus outside of time." This is from Aquinas.
The problem is a real causal chain involves complex events being caused by less complex events, all the way to an ideal 'simple' cause. When you interject God into this chain you get down to "almost simple" and then you inject an infinitely complex being into the mix, which starts the causal chain over again in a massive event of complexity.
You do this by special pleading. All events need a cause, therefore something exists outside of these events, that is God.
The problem is by creating a being outside the causal chain you invalidate the need for such a being, since now you have an event that doesn't need a cause, so, "all events need a cause" is broken. Essentially, what caused God? And if a causeless event is needed, it is more likely the event is infinitely simple rather than infinitely complex, or "why call the event God?"
The problem of evil is only a problem for the modern traits given to God, namely omnipotence, omniscience and omnibenevolence. They don’t play into cosmological arguments.
