Can a non-instant or sorcery ever have flashback?
That's correct.

It's currently impossible for a permanent card to gain Flashback[1], but let's say it's possible. Instead of being put on the battlefield on resolution, the card would be exiled instead.


  “Flashback [cost]” means “You may cast this card from your graveyard by paying [cost] rather than paying its mana cost” and “If the flashback cost was paid, exile this card instead of putting it anywhere else any time it would leave the stack.” 





Flashback can only be granted to instants and sorceries, and it would be too messed up for a card with a permanent type to gain one of those types.