Modern discussions of Kantian transcendental arguments were framed by Stroud's strong critique of them in his 1968 paper. Good sources on the nature of these arguments, Kantian and modern, and subsequent discussions of them are Hintikka, Transcendental Arguments: Genuine and Spurious (1972) and Pihlström, Peircian Scholastic Realism and Transcendental Arguments (1998). There is a recent volume Transcendental Arguments in Moral Theory (2017) on that subject specifically.
Briefly, the answers to both questions in the post are negative. As traditionally understood, Kantian transcendental arguments are plainly deductive: if P (phenomenon) is possible then C (condition of possibility), P (is actual), therefore C. Hintikka argues that this scheme (which he ascribes to Gram and Strawson) is much broader than what Kant intended:
"... According to the typical current interpretation, any inference from a successful conceptual practice (use of concepts a priori) to its presuppositions is a transcendental argument. I take it that such an argument is transcendental in Kant's sense only if the conclusion is an assertion concerning the process of our coming to know the objects of the knowledge in question."
But let us put that to one side. A deeper criticism, raised by Stroud, was that the minor premise, the assertion of P's actuality, is in principle impossible to establish a priori. Hence, such arguments are inherently incapable to "answer the skeptic" in the way Kant ostensibly intended (in the Refutation of Idealism, for example):"the sceptic can always very plausibly insist that it is enough to make language possible if we believe that S is true, or that it looks for all the world as if it is, but that S needn’t actually be true". After Stroud, pure Kantian transcendental arguments fell out of favor, and even more sympathetic authors, like Palmer and Taylor, acknowledge their much deflated effect.
The tendency we see in modern employment of transcendental-like arguments is to relax the assertion of P, and reinterpret the argument as abductive instead. It becomes an inference to the best explanation from a plausibly surmised P (objective world, our knowledge of it, our morality, etc.) to explanatory C, not unlike from empirical observations to high level scientific theories. The logical form would be: C is the "best" explanation of how P can happen, P (is plausibly in evidence), therefore (plausibly) C. Pihlström explicitly advocates such "pragmatization" or "naturalization" of transcendental arguments, the idea he traces back to Peirce, the coiner of "abduction".
"We should not forget that Kant himself, the father of transcendental philosophy, took something contingent and factual for granted... that we have experiences of an objective world and then tried to determine the necessary conditions of the possibility of such experiences... Kant did not ask whether experiencing an objective, structured world of appearances is possible; he asked (transcendentally) how it is possible.
[...] By means of transcendental arguments, we are able to critically investigate the conceptual network we take for granted in our discursive practices. These arguments need not "prove" anything to the skeptic who desperately wants to stand outside those practices... As soon as our naturalism is "retranscendentalized", we may drop the duality between a general, high-level abductive hypothesis, which explains certain phenomena, and a transcendental principle, which determines the conditions of the possibility of those phenomena. This requires that we loosen the requirement of the aprioricity of transcendental principles.
[...] I am not denying that a purely logical gap can still be seen between transcendental and abductive argumentation. The difference is that (TA) is logically valid... whereas (AA) is invalid (but can be regarded as a non-deductive inference typically employed in science). It would be a bad idea to formulate the naturalization thesis so that it would destroy this logical difference. The thesis can, rather, be understood as an insistence on the irrelevance of that difference in our practice of legitimating such features of our experience as cognition, science, language, etc., features often defended by means of transcendental arguments."
With this in mind, let us examine typical moral arguments for God's existence. We quickly find their "abductization" on full display, so they are not transcendental arguments in Kant's exacting a priori sense. As indeed they can not be, considering the emotive, error theory or evolutionary counters to the supposed evidence for moral facts needed to get them off the ground. Here is a sampler:
"The apparent connection between morality and religion appears to many people to support the claim that moral truths require a religious foundation, or can best be explained by God’s existence, or some qualities or actions of God.
[...] Theoretical moral arguments for God’s existence can be understood as variations on the following template: 1) There are objective moral facts. 2) God provides the best explanation of the existence of objective moral facts. 3) Therefore, (probably) God exists."
