Special Summoning After the Seal of Orichalcos

Question

My question is this, after you activate the seal can you Special summon monsters from other places?

The Seal of Orichalcos destroys all monsters that you had special summoned when it activates and It specifically calls out that you can't from your extra deck.

When this card is activated: Destroy all Special Summoned monsters you control. You cannot Special Summon monsters from your Extra Deck. 

However once the card has been out on the field can you Special summon again or is the effect continuous and would destroy any addition Special summoned monster?

Answer 1

My question is this, after you activate the seal can you Special summon monsters from other places?

Yes. The card is being explicit by stating that you can't special summon "from your extra deck".

You are free to Special summon from other places (hand, deck, graveyard, banish).

However once the card has been out on the field can you Special summon again or is the effect continuous and would destroy any addition Special summoned monster?

Once the card leaves the field all of it's effects stop working. It is a continuous effect, but precisely because of that is that it's effects only last while face up on the field. From the wiki on Continuous effects we have, emphasis mine:

Continuous Effects (Japanese: 永続効果 Eizoku Kōka) are a type of monster effect that applies while the monster [or card] remains face-up on the field.

Answer 2

You're mixing 2 effects.

The Seal of Orichalcos:

All monsters you control gain 500 ATK. Once per turn, this card cannot be destroyed by card effects. While you control 2 or more face-up Attack Position monsters, your opponent cannot target your monster(s) with the lowest ATK for an attack. If this card is activated: Destroy all Special Summoned monsters you control. You cannot Special Summon monsters from the Extra Deck. You can only activate "The Seal of Orichalcos" once per Duel.

About the destroy effect:

Note "If this card is activated: Destroy all Special Summoned monsters you control.". So the destroy effect happens only once, which is when you activate the card.

About the prevention of special summon:

It prevents you from summoning from Extra deck only, so as @DarkCygnus mentioned, you can still special summon from other places (hand, deck, graveyard, banished). These ones can be special summoned AND won't be destroyed (because the destroy effect happens only when you activate Seal of Orichalcos)