Is it a sin for the supplicant to sometimes feel despair and have misgivings, while continuing to supplicate and being certain that Allah will respond, but the answer is delayed?

You must have a good opinion of Allah and continue to supplicate, for supplication is an act of worship. The answer to your supplication may be the fulfillment of your request, or the warding off of evil, or the storing of good for you in the Hereafter. In the Hadith: "The supplication of anyone of you is granted as long as he does not supplicate for a sin or for the severing of kinship ties, and as long as he does not become impatient." In another Hadith: "No Muslim supplicates to Allah with a supplication in which there is no sin or severing of kinship ties, except that Allah gives him one of three things: either He hastens his supplication for him, or He stores it for him in the Hereafter, or He averts from him an evil equivalent to it." Supplicate to Allah with certainty and do not despair, for Allah knows what is best for you, and know that Allah does not answer a supplication from a heedless, distracted heart. As for satanic whispers, there is no sin in them unless you intend them or give in to them.