If a supplication comes to my mind, does that mean Allah inspired me with it so He may answer me, and will Allah answer a supplication I made last Ramadan before the coming Ramadan?

It cannot be definitively stated that a supplication made in the previous Ramadan will be answered in the coming Ramadan, or before it, or after it. There is no necessary correlation between the answering of a supplication and the fulfillment of the exact desired outcome, for Allah may answer supplications in various ways. This is indicated by the Hadith of the Prophet Muhammad, peace and blessings be upon him: "There is no Muslim who supplicates with a supplication in which there is no sin or severing of kinship ties, but Allah will give him one of three things: either He will hasten his supplication for him, or He will store it up for him in the Hereafter, or He will avert evil from him equivalent to it." Furthermore, there are conditions and causes for the answering of supplications that should be observed.