According to Islamic theology, will Allaah answer my prayers even if I sin consciously; disbelieve in heaven and hell, the Prophethood of Muhammad, the divinity of the Quran, and a host of other Islamic things; and do not pray the Islamic way?

All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His alave and Messenger.

First of all, we would like to draw your attention to the fact that even if the question is hypothetical and you are not the one who "sins consciously and disbelieves" in such matters, it is better in such a case to use the third person pronoun in your question. For instance, the Muslim may ask, "Would Allah accept the prayers of someone who sins consciously and disbelieves in such-and-such?" The questioner should not attribute such sinful actions to himself.

The most likely correct view in this regard is that Allah may accept the supplications of even the disbelievers. Shaykhul-Islaam Ibn Taymiyyah said:

"All people ask Allah, believers and disbelievers alike, and He may answer the supplications of the disbelievers. The disbelievers ask Allah for provisions, and He grants them provisions and water. If harm touches them at sea, they will turn to no one except Him. Then, when He brings them safely to land, they turn away and man is ever ungrateful. Allah, The Exalted, says (what means): {And when adversity touches you at sea, lost are (all) those you invoke except for Him. But when He delivers you to the land, you turn away (from Him). And ever is man ungrateful.} [Quran 17: 67]"

Ibn Al-Qayyim said, "Not everyone whose supplication Allah answers has attained His pleasure or love, nor is He pleased with his actions. He may answer the prayers of the righteous and evildoers, believers and disbelievers." For more benefit, please refer to fatwa 259408.

If a person disbelieves in Paradise, Hellfire, and the Prophethood of Muhammad, sallallaahu ‘alayhi wa sallam, then performing the prayers would not do him any good, even if he performed them properly as prescribed. Allah, The Exalted, says (what means): {And what prevents their expenditures from being accepted from them but that they have disbelieved in Allah and in His Messenger and that they come not to prayer except while they are lazy and that they do not spend except while they are unwilling.} [Quran 9:54]

Allah knows best.