"He did not bank with the old family banks."
1. Does the first 'bank' mean 'deal'? = He had
no account with those banks.
2. Is there a special bank called the 'family bank'?
Or is it a bank his family used to use before?
3. Does this 'old' mean 'previous' or something like that?
= the family banks he had used before.
"old family banks" - these are the private banks, such as Coutts in London or Rothschilds, as opposed to banks which are now public companies. They have a connotation of stuffiness and being restrictive.