As long as you created the account, and registered the key (and guest key) then all you need is a copy of the client, log in and play. (and the keys becomes all but useless in 90% of situations after that)
You can clearly access the account, so im not sure what your actually after, in that regard.

As for the move to new pc, not sure how or if FFXIV can be moved to a clean install.
Once as my SSD arrives and I've had time i'll be doing a clean install so i'll post back if you can move it across.
But you can get the basic client onto a new machine without the discs.
( i just doubt I can say how on official forums lol)