Does God answer the prayer of one who asks for dominion over the wind, or a kingdom like that of Solomon (peace be upon him), in our present age where miracles may not occur, and is such a supplication considered an innovation (bid‘ah)?

All that you have mentioned falls under the category of aggression in supplication (Du'a), which is forbidden in His (the Most High's) saying: {Call upon your Lord humbly and secretly. Indeed, He does not like aggressors}. Aggression in supplication occurs by asking for what is impermissible to ask for, such as assistance in prohibited acts, or asking for what Allah does not do, or asking for positions that are not befitting for Him, like the ranks of the Prophets.

It is incumbent upon the Muslim to supplicate to Allah with comprehensive supplications that encompass the good of this world and the Hereafter. Aisha (may Allah be pleased with her) used to say: "The Messenger of Allah (peace and blessings be upon him) used to prefer comprehensive supplications and abandon what was other than that."