I know I appreciate corrections to my posts on the Spanish/English forum, so I hope you won't mind my corrections. You can say "Thank you for answering quickly" or "Thank you for your quick (or prompt) answer."
"Thanks a lot for answering quickly" or "Thanks a lot for your quick answer" are fine too.
In offices that still use paper forms instead of electronic ones, one may have two containers on one's desk. Paperwork coming in that has not yet been dealt with is placed into the "in box" or "in basket" (the shape and material of the container determines which word would be used), while paperwork that has been attended to and is being dispatched elsewhere is would then be placed in the "out box" or "out basket".
Computer terminology has picked up the term "inbox" for the place one finds new, unread messages.