The text is, "When you are dealt a face-up Damage Card with the PILOT Trait, discard it immediately without resolving its effect".

The Damage Deck has 8 PILOT Trait cards of the 33 cards, roughly one-fourth of the deck are PILOT Traits.

My first question... Does this mean that 1/4 of the time a ship is dealt a Critical Hit, it can not only avoid the PILOT effects, but the hit on the ship's hull itself because the Damage Card is discarded, not turned face-down with no text to direct dealing another Damage Card for the Critical Hit?

My second question... Does this further mean that effects that turn an existing Damage Card assigned to a ship face-up are applied, even if it has a PILOT Trait? After all, in this case, the Damage Card is not dealt to the ship. Further, if in this case the text of the Determination Upgrade Card is applied, would not the effect be to "heal" a damaged ship one hull point?

The text is, "When you are dealt a face-up Damage Card with the PILOT Trait, discard it immediately without resolving its effect".

The Damage Deck has 8 PILOT Trait cards of the 33 cards, roughly one-fourth of the deck are PILOT Traits.

My first question... Does this mean that 1/4 of the time a ship is dealt a Critical Hit, it can not only avoid the PILOT effects, but the hit on the ship's hull itself because the Damage Card is discarded, not turned face-down with no text to direct dealing another Damage Card for the Critical Hit?

Correct. Determination negates the damage.

My second question... Does this further mean that effects that turn an existing Damage Card assigned to a ship face-up are applied, even if it has a PILOT Trait? After all, in this case, the Damage Card is not dealt to the ship.

Correct as well. Determination only applies if you are dealt a card. If anything happens to an existing card (saboteur, rexler) determination does not trigger.

As dvor mentions it looks like you have things correct.

When you have Determination:

1. Any face-up card you are dealt with the pilot trait is immediately discarded. It's like it never happened.

2. Any other time Determination does nothing.