To be honest the Post Office is well capable of doing all of this. But Monzo will have to pay to arrange this. I would suggest Monzo deducted 1 or 2£ from the deposit to cover the costs.
What most people don’t know is that almost all banks allow you to deposit cash straight away into your account by handing the card to the cashier and swipping it. Some post office cashiers get a little bit confused but it is possible.
Re the cheques, it would have no value. What the PO does is to mail them for free to your bank. Some of them will even give you a bank branded envelope that you can just put in any mailbox whenever you want.
This comes from my experience of when I lived in rural Scotland and the post office where our only way to bank.