There is no dispute amongst the followers of the Ahlul-bait (peace be upon them) that the Prophets and the Imams are sinless, in such a way that they do not commit what opposes the religion. Our scholars have explained in their books of beliefs the evidences to prove this concept. There is no doubt that the Prophet and the Imam are appointed by Almighty Allah. He has ordered people to follow them. So, if it is possible for them to commit sins, they might order people to do so as well and people have to follow them. So their orders would not be trusted to be from Almighty Allah, and He will not order us to follow somebody who can misguide us. Similarly, they are to be taken by their followers as role models. If they were to commit sins, others will be encouraged to do the same.
Yes, it is permissible to live like that as marriage is highly recommended and not compulsory, although Islam strongly recommends it in many narrations, like when the Messenger of Allah (peace be upon him) said: "There is no deed in Islam that Almighty Allah loves more than marriage."