If you delete a character, does its designs stay?
Solution 1:
My educated guess is no, they will not stay. According to this forum thread if you overwrite a design everything that uses that design will be changed to the new design. I would imagine deleting a character would have the same effect of overwriting those designs with 'nothing'.
After testing it I can confirm that any designs placed by the deleted character are removed.
This is still true even if another character has the same design in the same slot. I had the temporary character place a design that both characters happen to have and it still got deleted.