What happens if you have a client who wants you to market something grey hat/black hat on FB and gives you access to their account through your own? In other words, you are marketing their product using your own account but with the client being billed (standard fare stuff.) If FB disapproves ads or gives warnings because of those ads, does it affect your account as well as the client's account?
Ask yourself why the client is not letting you use their FB account directly instead ... I think you already know the answer to this one.
What do you mean here - the wording is ambiguous.
Has the client given you access to their account or permission for your account to advertise their page/app?
These are very different situations. In the former case everything goes through their account. In the latter everything goes through yours and you would have to invoice them.
If they want shady stuff, I wouldn't run it on your account.