Is one who believes, for a specific moment, in the whispering doubt concerning the existence of Allah an infidel (kafir), and will his repentance be accepted?

Your dislike of these whisperings is a sign of the sincerity of your faith. You have not exited Islam because of them; rather, they are proof of the soundness of your conviction and the strength of your faith, as the Prophet, peace and blessings be upon him, said: "That is pure faith." The cure for these whisperings is to struggle against them and turn away from them. With sincere determination and strong will, Allah will bless you with recovery, and you will be rewarded for your struggle against them.