The LOOP

You can perform multiple LOOPs in one turn but each costs 1 more Energy than the previous (1 for the first, 2 for the second, 3 for the third, etc.); each LOOP spends 1 Energy from your Era and readies all cards in your hand that share one chosen Elementary Dimension (you may never choose the Black Hole Dimension). (p.7)

Any Rift you remove is returned straight to the pool (not placed on an Era), so that particular Rift can never cause a Vortex; generally removed Rifts go back to the pool. (p.11)

If a card is made available on an Era that becomes a Vortex, the card’s effects fully resolve first and it is then destroyed by the Vortex. (p.12, p.6)

No — you may only archive a Sabotage tile during your Archive phase and only if you are on that tile’s Era; archiving also triggers the Attendance Bonus. (p.9, p.10)

Yes — destroying a Duplicate on the Era where Dr Foo is located counts toward the corresponding Sabotage mission and lets you place the appropriate damage cube(s). (example and explanation: p.8; missions summary: p.10)

For your Agent deck, shuffle your Discard pile to form a new Draw pile and then finish resolving the draw; for the Huge Deck, if it is ever empty you reshuffle the Destroyed Cards pile to form a new Huge Deck. (p.5, p.11)