I would like to ask about Muslims who make their income from a Haram source (for example, selling alcohol in their business). A Hadeeth says what is in the meaning that when such a person asks Allah for His help, Allah does not respond to him because his source of income is Haram.
My questions are:
(1) These Muslims acknowledge that they are doing something Haram but that they are weak in front of their desires. Does the fact that their source of income is Haram make them a Kaffer (non-believer) or do they remain as Muslims?
(2) The Hadeeth mentions that Allah does not answer them when they ask him for help. And of course their Zakat and Hajj are not acceptable because their money is from a Haram sources. Is their Salah, Fasting and reading the Qur'an acceptable to Allah or are all their good actions not acceptable according to Islam?

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 is His slave and Messenger.

People who claim to be Muslims cannot be suspected of disbelief for any sin they commit unless they say that what is forbidden in the Sharee’ah is lawful or they worship anything other than Allah.

So, those who deal in Riba (interest and/or usury) and alcohol to please themselves in this worldly life, are treated as Muslims. If any of them dies without repenting, Allah will do with him as He pleases. He might forgive him from His Mercy and Compassion or He might punish him proportionally for his sins. If Allah punishes this person in Hellfire, he will not stay there forever because he died while believing in Allah and His Prophet .

It is reported in a sound Hadeeth that Prophet said: “Go and take out (of Hell) anyone in whose heart you find faith equal to the weight of an atom (or a mustard seed).”

As for your second question: No doubt, Allah only accepts what is pure, so He does not accept ill-gotten money or its Zakat. The supplication of a person who consumes ill-gotten money is not accepted by Allah as narrated in the Hadeeth mentioned in your question.

As for performing Hajj from such money, there is no consensus among the scholars over this issue. Most of them believe that such a Hajj suffices although he has committed a sin. But Imam Ahmad believes that this Hajj does not suffice him for the Hadeeth in which the Prophet said: “O people, Allah is Good and He, therefore, accepts only that which is good”. [Muslim]

As for the prayer, reading the Quran, fasting and all other physical forms of worship, they are valid and he is not asked to perform them again.

Allah knows best.