If "Barrier Seal" is destroyed by a spell, it will be destroyed and put into a removed area before that spell would leave the chase. Thus, the spell will go to the graveyard as normal.
While "Barrier Seal" is applying its effect, abilities that trigger when a card is put into a graveyard do not trigger, as the cards are moved directly to the removed area from their original zone.
An ability that triggers "when this card is discarded" or a spell/ability with an additional cost of discarding a card will still work normally while "Barrier Seal" is applying.
Ex."Seth, the Arbiter" has an ability that triggers "when your opponent discards a card". This will trigger even if your opponent discards a card and it is put into a removed area because of "Blazing Seal". However, if "Charlotte's Protector" is put into a removed area when discarded by "Charlotte, Determined Girl", it will not trigger, because its trigger is "When this card is put into a graveyard from your hand", which it was not.