when chewbacca takes a critical hit as the result of draw their fire is the damage card face up or face down? I am confused due to the faq involving the difference between suffering and dealing damage.

Face down, that's why people like running it with chewie.

The technical answer is that the card is dealt face up, but chewbie flips it immediately facedown.

Suffering damage implies a process to check for the presence of shield tokens on the ship, to later assigning damage cards according to uncancelled dice results (IE: Weapon attacks) or other effects (IE: Ion effects, Darth Vader crew) if there aren't any shield tokens present.

However, when an effect calls you to directly 'deal a card' the above process is bypassed, without checking for shields, and the ship recieves the card as stated by the given effect (IE: Proton Bomb).

Draw their fire transfers the kaboom result before the damage card is dealt so the card is still being dealt to Chewie, therefore his ability works.

As Forgettenlore mentions, DtF transfers the "suffering" of the [crit] result. When it finally gets to resolve it either removes a shield token or then deals a face-up damage card which Chewie's ability immediately turns down.

In a related issue that could come up in the future (Rebel Aces) a ship could have DtF and Determination. While this combo would be legal you will have to decide to use DtF before you'd have any idea what the face-up card would be should you take it. This means you can use the combo to ONLY take pilot-critical which would then get discarded by Determination which is how one would want, almost even need, to work on an A-Wing. Now if you could get that combo on Chewie....

I should also point out that Rexler Brath works after Chewy flips the cards down. Here's the order of operations:

Chewy suffer an uncanceled hit and critical hit result from Rexlar

Chewy checks to see if any shield tokens remain. For purposes of this example, he does not.

Chewbacca suffers face down damage.

Chewbacca suffers one face up damage. Both players get to see it.

Chewbacca flips that faceup card facedown.

Rexler may then spend his focus token to flip BOTH cards face up. Chewy then suffers these effects.

It is important to note that Rexler can choose to force Chewy to suffer that crit after he has seen it. If he only gets one crit through, for instance, he may choose to flip it if it's a damaging crit such as ignore pilot ability or blinded pilot, but not do so if it has no impact, such as discard secondary weapon.

Can you use this in combo with the falcon title and burn an evade to get rid of the hit all together?

Can you use this in combo with the falcon title and burn an evade to get rid of the hit all together?

No, the evade token can only be spent in the modify defence dice step, and that has already been passed by when the crit is pulled with DTF (which happens after all the dice are compared and the crit is not canceled out).