Archimago

When I cast one of the banished spells, is that placed back in the cemetery?

No. There's no way back from being banished, which overrides the part of resolution that would place the card in the cemetery.

When I use the Archimago's ability to banish 3 magic spells and cast 1, do I still have to pay mana and meet threshold?

Yes.

When using this ability, must the Archimago be the caster?

No, any of your spellcasters can cast the copied spell.

What happens if I pay the cost of banishing three spells, but I cannot actually cast any of those three?

If you cannot cast any of those three spells, they remain banished and nothing else happens. You did not cast a spell. (If you can cast one of the three, you must.)