Is there a deadline date where the code we received will no longer work?
We don't have one, but at some point we will move onto other folks waiting on the list and there might be price changes. Let me know if you run into any issues with an expired code.
In case that website server is not working or permanently down, that means that Bitcoin note can't complete authentication, but can they still be used without that or not?
Oh, and you forgot to say when can we expect first deliveries for ordered bitcoin notes?
There is a sliding scale. If we are temporarily down, you can "offline" verify and see that the multisig is still loaded (assuming you can get data from the chain). If we are permanently down then you need to wait until the expiration ("claim before") date printed on the note.
First deliveries are expected late August.