We are in Philosophy.SE here, not in Religion.SE or Christianity.SE. So the best we can do is use what tools Philosophy offers and try to tackle the issue purely on a formal level.
In this case this means to give a definition for every term, translate them into the realm of logic, and accept whatever outcome.
Your five statements are very clear, well defined and do not require any definitions from other sources. They consist of the first two givens ("axioms" or definitions), and three steps ending in a contradiction. The logic instrument used is reductio ad absurdum or proof by contradiction: the argument is designed to show that all the axioms together cannot be true.
So, to repeat: your argument proves, by logic, that not all of these statements are true; at least one must be false:
- God can do anything at all
- God wants that everything that exists is good
- Evil exists (equivalently: God has created Evil -or- Evil has somehow crept in and God can not get rid of it) and Evil is not good
(These are the same as the first few bullet points from your question, just slightly reformulated to be more rigorous or explicit or to use simpler terms.)
((Note that we use the term "exist" multiple times - this is a very complicated word whose meaning is by no means clear; but let's not go there now.))
The reductio ad absurdum proves that at least one of these statements is true:
- God cannot do some things (i.e., remove Evil, or create an universe without Evil)
- -or- God does not want to do good in all cases (i.e., God added Evil for nefarious reasons; or refuses to remove it although he could)
- -or- Evil does not exist (or equivalently: we just don't "get it" and what we claim to be Evil is indeed not)
The argument does not say more, or less, from a philosophical point of view. It would need further axioms to deduce more.
From the standpoint of logic there is not much more to say here. There is simply not more fodder to work with. No further conclusion can be drawn without more context.
--- Logic / philosophy ends here, but people in the real world of course continue the argument anyways: ---
Anything further would be pure speculation. Obviously, if you define God to be all-powerful and benevolent, then assuming the first or second result would show that God does not exist (at least not matching the Christian definition).
You could still assume that the solution is the third "Evil does not exist". The typical argument from Christians I hear is that "God moves in mysterious ways", i.e. what seems like Evil to us is, in fact, not. So taking this "out" will avoid the conclusion that God does not exist.
But of course this would not allow you to conclude that God does exist - it simply invalidates the whole argument (by removing one of the premises) without any conclusion at all.
