They would just be deck codes, which would take up less than a Kb I would imagine, and since all of your cards are stored, a few deck codes wouldn't put any strain on the servers memory capacity.
are you referring to simply a temporary cache that gets removed when you close the client?
no, permanent, since it would require very little, next to no memory usage to store the codes, a player could store up to 3 (i think 3 is a nice number) of their favourite or most used decks, and this way can swap between them at the push of a button.
It is possible to send PMs to yourself with deck codes.
why would i pm myself? i store my deck codes in a notepad file.