What happends to your stash if your have only 1 hardcore Hero and he dies ?
is this a problem ? do you need at least 2 heros ?
Stash is account bound so your items are safe. Same thing if you delete all your characters and the create new ones. The stash items remain for the account.
what Purkeri said. Paragons, and everything in your stash remain there. Inventory items though get deleted, so you must learn how to move everything in your stash when pushing.