Q. If a book from the remote storage building is checked out already, should I request one through ILL?

Generally, ILL doesn't borrow things we already own.

Storage items have a short load period. If you use the same Request method you'd use to get the item retrieved from storage, it will place a hold on the currently checked out item. That prevents it from being renewed, and as soon as it is checked in, it will be held for you at the circulation desk instead of being sent back to the storage building. You'll get an email as soon as it is ready.