Wny would his job need a utility bill or tenancy agreement? What did he say when you asked him about this? This is the type of information he would need to open a bank account and sometimes for a school place. Tell him if he needs these documents, he can't use Airbnb and will need to go through an estate agents or rental agency and they will issue him with a assured shorthold tenancy, carry out credit check him etc; but that many will not take pets.
If you want to accept the changes he has suggested, you need to go into his booking and add the extra two days and add in the wife and child joining him three months earlier. Let him know that he must accept these changes before you are able to let him check in. And get him to confirm about whether the dogs are coming.
Tell him the only thing that Airbnb provides him with is his booking confirmation which he can give to his new employer.
How long are your guests staying for? What sort of discussions did you have before accepting a long term booking? Do you feel your space is suitable for a long term booking? What sort of bed/equipment are you providing for the child?
Personally, I would also have turned down the booking as I would be uncomfortable about potential damage caused by three dogs and a small child on a long term booking (you do know that pets aren't covered under the Airbnb guarantee don't you) and because I wouldn't be confident that the space would work as a long term space.
If I was persuaded to accept the booking, I would have insisted in costing in at least monthly cleans to keep an eye on the condition of the listing. And I would probably have suggested a shorter stay through Airbnb and then a proper tenancy agreeement for a longer stay if things had worked out.
You shouldn't be discussing change of arrangements by phone, this should all be done on the Airbnb messaging platform so there is a record. Make sure you message him back and say further to our conversation this afternoon XXX and XXXX. Ask him to contact you via the messaging platform if he wants to discuss further changes.
I can see why you are uncomfortable but it is difficult to advise based on the information provided in your initial post.
If you are considering cancelling, then you need to do this asap so the guest can find alternative accommodation.
