I haven't played Star Trek Frontiers but based on my familiarity with Mage Knight and having read the rulebook I'll throw in my two cents.
My ship got attacked with a 4 Pulse Torpedo Attack. I don't have any Pulse Torpedo Shields, but I was able to generate 5 (normal) Shields.
1) I halve my shields rounded down, and block 2 of the damage?
My captain is level 5-6, with a value of 3 Shields.
2) I take a damage card, halve his 3 shields, and block a single damage. I do this again to block the other damage. So I have taken 2 damage cards?
At the point of assigning damage you are past the block phase and no longer care about shields and whether they are inefficient.
You have the rest of the process correct in that you take damage then reduce by the Captain's defense value (which aren't shields). This leaves you with no left over damage.
This enemy also happens to be a Borg Cube who has Antimatter Weapon and Biogenic Weapon.
3) The 2 initial damage that got past my ship's shields is doubled to 4 damage. The final verdict is that I take 4 damage cards in my hand, and 4 damage cards in my discard pile?
Modifying the damage as above your logic is correct. The damage is doubled from one to two from Antimatter and additionally take two damage to discard from Biogenic.