1. “First, why would it be impossible to conceive of God's non existence? ” «Let us assume that the subject of all perfections does not exist: then non-existence is a perfection, and in the subject of said perfection said attribute must be expressed without any limits, which would imply that not only does it not exist outside of the mind, but neither does it exist as an idea in the mind. Therefore God does not exist as an idea in the mind. But God does exist as an idea in the mind, therefore the assumption that the subject of all perfections has the perfection of not-existing led us to a contradiction. »
2. “God is morally perfect. But one might think that a God who is morally perfect would not be responsible for having created anything less than perfect - such as this world - and so conclude that God does not exist, and conclude this 'from' his moral perfection.” That is a fair objection. It may be, however, that only God (if he exists) knows the reason why it is morally better to create something imperfect rather than not to (see Leibniz's doctrine of compossibles and also his proposed solution of the problem of evil) (Notice I said «it may be» that way, not that God exists and it is in fact like that).
3. «But even if that kind of reflection is confused and properly conceiving of God does involve conceiving of him existing - which seems quite dubious - that would not conclusively establish his existence, as it is possible to conceive of things that are impossible. For instance, when I mistakenly think that 4 x 93 is 374 I am conceiving of something impossible (for it is not possible for 4 x 93 to equal that figure).» That's an interesting argument. However, you can only say that it is impossible after reflecting upon it. It may be impossible that the subject of all perfections exists, but it is not enough to say that it is, as you say. Whether it is so is still a matter of debate.
Perhaps I can give a clearer picture of the argument with this passage concerning Leibniz:
«(...) Leibniz wrote out a proof that the idea of God is possible (...) This proof defines God as the most perfect Being, i.e., as the subject of all perfections, and a perfection is defines as a "simple quality which is positive and absolute, and expresses without any limits whatever it does express." Leibniz easily proves that no two perfections, as above defined, can be incompatible. He concludes: "There is, therefore, or there can be conceived, a subject of all perfections, or most perfect Being. Whence it follows also that He exists, for existence is among the number of the perfections."» (Source: Bertrand Russell, History of Western Philosophy).
4. «the problem is that God is not a necessary existent, for God is omnipotent and thus can do anything, including taking himself out of existence.» If you define omnipotence like that, then that is obviously correct. But in order to inquire epistemologically into the most coherent and logically viable notion of God (and also to abide to the principle of charity), we should look at all notions of omnipotence that seem as plausible as possible. For instance, one may define omnipotence as the ability to do anything except what is contrary to the laws of logic. That God cannot kill himself may only confirm his omnipotence, since some also define him as immutable, and dying would imply that he could change (if he existed) which is impossible by definition.
5. “Thus those who - like me - believe in God, must conclude that nothing exists of necessity.” See my response to 1, it is a reductio ad absurdum.