The answer is basically `Yes` to all your points.
But, the bank may well want to confirm with your wife (as the account holder) that you have been authorised to withdraw cash on a crossed cheque rather than have it put into an account with your name. As a result you may either need a letter of confirmation with the subject cheque number, amount, your full name and your wife`s signature plus your own personal identity documents (passport or similar) or for the bank to be able to `phone your wife, confirm they are speaking with the account holder and get her approval that way. The end result will also depend on the value of the cheque, the higher the value the more difficulty.
Generally, it is easier to have your own account from which you can draw cash by cheque with much less bother.