Hajj-e-Badal

Doing Hajj-e-Badal for a person who has dementia

Q: Hajj was fard on a person, but he delayed it until he became old and lost his senses due to severe dementia. Is it valid for his family to fulfill the fard for him by: 

1. Bringing him along for Hajj and making the intention of ihram etc for him 
2. Doing hajj e-badal for him during his lifetime (note that he is not in his senses to command this, but it will be fully on their own initiative) 
3. Doing hajj e-badal for him during after his death 

Please provide the ruling for each of the three scenarios. 

What should be done with the excess money that remains after performing Hajj-e-Badl?

Q: In the case where the deceased did not perform his fardh hajj, and before passing away, he appointed someone to perform his fardh hajj (Hajj-e-Badl) from one third of his wealth, then after performing the Hajj-e-Badl, if any money remains from the one third of the estate, what should be done with the remainder money?

Will Hajj-e-Badl be compulsory if one delayed in performing Hajj and then passed away during the journey of Hajj?

Q: Explain the ruling of a person who delayed in performing hajj after it became fardh upon him and only set out for hajj after one or a few years, and then passed away during the journey of hajj. Does Hajj-e-Badl have to be performed on his behalf?