There is a vast array of logical objections to theism. Many objections aren't included here. New objections will no doubt emerge. Do any succeed?
We have seen how, with some ingenuity, it is always possible to find a way round an internal and external logical objection, by, say, tweaking your definition of God, finding some fault in the logic, or, in the case of external objections, giving up one of your other beliefs. Does this mean that all logical objections fail?
In so far as such objections force theists to revise their position, no.
What these objections target are very specific conceptions of God (sometimes in combination with other beliefs) and, in many cases, the theism that involves those very specific conceptions must, on pain of contradiction, be abandoned (or else the other beliefs abandoned). The target theistic belief or belief-combination often is successfully refuted.
Theists sometimes acknowledge this. However, other varieties of theism, very loosely understood, always remain on the table. So the theist can always switch to one of those other varieties. For example a theist struck by logical objections to a personal God may, in response, switch to a non-personal conception of God. Indeed, the theist may insist that one of the other varieties was always what they had in mind. They've just been misunderstood, and/or themselves unclear.
From the perspective of the theist, this strategy of switching and adjusting belief in response to such objections constitutes progress in getting clear about what theism fundamentally involves. Such logical objections are helpful to theism, they say, by allow theists to clarify the nature of God.
From the perspective of many atheist critics, on the other hand, these same logical objections are in many cases a threat to theism, and the theistic strategy of switching, modifying, and/or abandoning beliefs in order to retain at least some sort of logically consistent theistic belief looks suspiciously like, not clarification, but rather a merry-go-round of evasion.
Which of these two perspectives is the more accurate is, I suspect, the fundamental question to press regarding logical (and indeed evidential) objections to theism.