Is the statement, "If God exists, then make me cough," considered an act of disbelief, and what is its effect on the past acts of worship and obedience of the person who uttered it?

This is one of the perils of delving into atheism and misguidance. The aforementioned statement does not, by itself, necessitate disbelief (kufr) unless there is doubt about the existence of Allah the Almighty; for doubt about the existence of Allah the Almighty is disbelief, except for one who is afflicted with obsessive doubts (musawwas). The prayer of an apostate establishes the ruling of Islam for him, because it includes the two declarations of faith (shahadatayn), unless his apostasy was due to denying an obligatory act (fard) or similar. In that case, his repentance is by uttering the two declarations of faith along with affirming what he had denied. You may refer to previous fatwas for further elaboration on the proofs of Allah Almighty's existence.