Can he divide his estate among his heirs when he is sick?

Firstly: 

Distribution of the estate can only come after the death of the one who is leaving it behind. This is the basic principle. The sick person may live for a while and need that wealth, or one of his children may die before him. 

Secondly: 

It is permissible for a man to divide his wealth among his heirs during his lifetime when he is in good health, so long as he does not intend to cause harm to some of the heirs, by withholding from some, or giving some less than they are entitled to for the purpose of harming them. 

This is regarded as a gift from him to his children, and he has to treat them fairly, giving each male twice as much as each female. 

Thirdly: 

Gifts given during one’s final illness come under the ruling on bequests and can only be given from one third of the wealth. But it is not permissible to make a bequest to an heir, because the Prophet (blessings and peace of Allah be upon him) said: “Allah has given each person who has rights his rights, and there is no bequest to an heir.”

(This hadeeth was classed as saheeh by al-Albaani in Saheeh Abi Dawood). 

The final illness is that in which it is thought most likely that the individual will die as a result of it. See: ash-Sharh al-Mumti‘, 11/101 

Fourthly: 

If the father is sick but it is not a final illness, then there is nothing wrong with asking him to divide the estate, especially as this will prevent disputes in the future. 

But if it is his final illness or an illness that is affecting his understanding and decision-making, such as Alzheimer’s, then no one has the right to dispose of his wealth or to ask him, when he is in that state, to dispose of any part of his wealth. 

Fifthly: 

With regard to what the father registered in the name of some of his sons, and that was a gift for no real reason, then this is favouring one child over another, which is haraam. He should take it back from them or give the rest of his children something similar to what he gave those sons. If he dies without setting the matter straight, then those sons have to give back what was registered in their names and put it with the rest of the estate so that it may be divided among all the heirs. 

If there was a reason for favouring these sons, such as if they were extremely poor or sick, then this is regarded as permissible by some scholars. 

With regard to the second question, it is not clear what was meant by it. Perhaps you meant that the father appointed a company to distribute the inheritance. This may be based on what was said above. If the father was healthy then there is nothing wrong with that. 

And Allah knows best.