The key thing here is that both Flashback and the "trap cost" are alternative costs, and only one alternative cost can be applied when casting a spell. The Flashback cost of 3UU is the only one that will allow you to cast the spell from the graveyard, so that's the one you have to use.
If my opponent searches his library and I have Archive Trap in the 'yard, can I Snap Trap and play it for 0?
Both Flashback and Archive Trap's first ability are alternative costs to cast Archive Trap, and you can only pay one alternative cost when casting a spell. As Flashback is the only cost that allows you to cast it from the graveyard, you need to pay 3UU to cast Archive Trap from the graveyard.
The only trap that works differently with Snapcaster Mage is Stone Idol Trap, because it has a cost reduction ("cost X less to cast") rather than an alternative cost ("you may pay X rather than..."). If you use Snapcaster on Stone Idol Trap, the Flashback cost will be reduced by the number of attacking creatures.
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Snapcaster mage
If my opponent searches his library and I have Archive Trap in the 'yard, can I Snap Trap and play it for 0?
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That said, in this specific case, the old answer is still correct.
Both Flashback and Archive Trap's first ability are alternative costs to cast Archive Trap, and you can only pay one alternative cost when casting a spell. As Flashback is the only cost that allows you to cast it from the graveyard, you need to pay 3UU to cast Archive Trap from the graveyard.
The only trap that works differently with Snapcaster Mage is Stone Idol Trap, because it has a cost reduction ("cost X less to cast") rather than an alternative cost ("you may pay X rather than..."). If you use Snapcaster on Stone Idol Trap, the Flashback cost will be reduced by the number of attacking creatures.