I dislike anyone saying that there must have been a god for the Big Bang to happen, for abiogenesis to occur, for evolution of complex structures (like the eye, cell or DNA) or to explain the reason why plain apes developed intelligence.
I agree with the two in bold. Abiogenesis cannot occur, that is proven definitively enough. Plain apes do not develop intelligence. Now adays, we just call them apes. The intelligent apes I assume you are referring to are humans, and therefore, distinctly different.
Incorrect. Science doesn't disprove things. It
can't disprove things. It is not possible to disprove something like abiogenesis, which is a hypothesis of life's origins in the course of millions or billions of years. Scientific method relies on proving hypotheses with experimentation, after which the hypothesis and the related means of predicting the outcome of experiments becomes a theory.
Same applies to God - science can't disprove him either.
Logical method can, though - you just need to ask yourself which is bigger, universe (by definition "everything that is") or God?
If God exists, he is by definition part of the universe which contains "everything there is". That means that if God is an individual entity in the Universe, logically it follows that the universe itself is mightier, bigger and awesomer than God, which kinda puts this God being on the same line with us, a sentient being in the universe.
Also, I don't really understand what you mean be "plain apes do not develope intelligence". When put into conditions where intelligence would be the primary selective pressure - whether naturally or by selective breeding - it's pretty certain that the intelligence of the test population would increase by time. However if other things like anatomical requirements for bigger brain would not be addressed (more dangerous childbirth and infancy, increased need for food etc. etc.) it would obviously at some point reach a limit where the appearance of the species would need to change for intelligence to grow further, and at some point you wouldn't be able to call them the same species any more - but the question is without much relevance because the required amount of generations for this kind of evolution is a lot longer than any research could conceivably last and thus we need to jsut use the nature as our laboratory and look at what kind of evolution is visible on our lifetime.
Such as rat populations developing resistance to strychnine. Or seagull populating spreading around the shores of Arctic Sea and when they met the original populations after going around the globe, the "new" and "old" seagull populations had deviated far enough that they didn't procreate between populations any more, because they didn't recognize each other to be of the same species any more. Whether that was due to changes in appearance or mating behaviour doesn't really matter; it is possible that at that point the two species (or sub-species) could have produced fertile offspring but the fact remains they didn't, which would have eventually led to the separation of the species into two.
In the case of the first one science can answer those questions to a good degree and can fill in enough details to make a claim that God did the rest seem rather like an attempt to shoehorn God in somewhere.
Why is it so unthinkable that God is the answer to a good degree and science is just the attempt to shoehorn what we assume we know in there? The Big-Bang theory and the theory of evolution are just that, theories. They have an observational scientific base going for them, of course, but remember too that abiogenesis was widely accepted because there was an observational aspect to it. The same may or may not be true for evolution or the big bang. They could be just as wrong.
Specifics first:
The big bang hypothesis is supported by the observations of expanding universe, and since the big bang hypothesis accurately offers an explanation to the observations, it can be called a theory and in addition it can be considered as the most accurate theory at present - which in science means that it is the most likely explanation aka probable truth.
Abiogenesis is a bit trickier thing. It hasn't been proven to happen experimentally due to obvious problems in replication of conditions and the timeframe for the expected emergence of life, which means it is a hypothesis rather than theory, but the thing that makes it more acceptable than God-answer in science is that it is definitely not inconceivable by other very well documented theories and observations - mainly physics, chemistry and observations of basic chemicals of life in various places in the universe. Abiogenesis is, in a way, an extension of chemical evolution. In favourable conditions, it is not inconceivable to think that some nucleotides or other molecules could form a self-replicating molecule, and when there's as much time as there has been in the universe (refer to the Big Bang theory, 13.7 billion years), then the probability of such an event happening at some point increases drastically.
Evolution on the other hand is a very well documented and proven accurate theory of the developement of species. Aside from fossil records, there's the huge amount of information used in genetic engineering and indeed even selective breeding that pretty much proves that it is indeed the genotype that defines the fenotype, and with sufficient change in the fenotype the species can evolve into different species (unable to produce fertile offspring).
Then the general answer. Goddidit is a lot less informative answer to things than some more in-depth answers. On a fundamental level, though, I personally find it easiest to simply call Universe a God without personality or overruling consciousness, and in that sense "goddidit" is as good an answer to things as saying that this is how things just happen in the universe.
Of course, considering Universe to be a blind god makes a lot more sense to me than a personal god for the aforementioned reasons (personal God < Universe).
What's worse though is when it has been explained and people insist on holding onto the "God did it" claim. It's as silly and insulting to those of us who do understand it as continuing to claim that God makes the sun rise every morning and insisting that all the astrophysicists have gotten it wrong.
Yes, it is. It's as silly and insulting to those of us who do understand that God made the sun, and therefore, has made it "rise" every morning as continuing to claim that all the religious people have gotten it wrong. Of course "rise" is just a spacially relative term in this case, but you get my point. "God did it" because He set in motion the things that astrophysicists try so hard to explain.
And of course there is the problem with "goddidit" that it hides everything else behind the question - where did God come from?
To me it all boils down to one thing. Which is more likely to emerge from nothingness - an all-powerful (with infinite powers) consciousness, or a finite universe without any intelligence of it's own, just a bunch of mechanics?
Not only that but there are no infinities in science, so it is unlikely that God would be infinitely powerful, even should he exist...*

*Smilie is there to announce that I realize how oxymoronic this sentence is...