I think I had it correctly the first time. Here are the relevant rules from the revised edition of ACtA, Book 1.
ACtA Book 1, p.16:
Dodge X: Some Ships (notably fighters or the legendary White Star) are exceptionally nimble and can make sudden, violent movements to thwart attack. Whenever the ship suffers a hit, roll one die. If the die equals or exceeds the Dodge score listed for the ship, all effects of the attack are ignored. The ship has literally dodged the attack.
ActA Book 1, p18:
Beam: Beam weapons have gained a legendary status in the galaxy and all races have tried to develop effective ones, although not all succeed. If an Attack Die from this weapon successfully strikes a target, immediately re-roll to attack again - the target number this time, however will be one higher than before. You may continue to re-roll every Attack Die that continues to hit the target, with a target number one higher than previously used, until you run out of Attack Dice.
ActA Book 1, p18:
Twin-Linked: These weapons are mounted in pairs or even quads, concentrating firepower available on each hardpoint or turret system. The hail of fire these weapon systems unleash is awesome to behold and very difficult to avoid. Any Attack Dice for this weapon that do not successfully strike their target may be re-rolled.
Reading these rules together seems to say:
If Beam Weapon:
1) Roll to hit with each die. "Whenever the ship suffers a hit, roll one die. If the die roll equals or exceeds the dodge score listed for the ship, all effects of the attack are ignored."
So if the ship dodges, the hit is no longer a hit and "all further effects of the attack are ignored." (I.e. It is as if the hit did not happen and you do not re-roll the beam's attack die to see if you inflict more damage per the beam rules.)
If the ship fails to dodge, roll the attack die again with a target number 1 higher. If the die hits. Repeat the dodge procedure. If the target dodges. All further effects of the hit are ignored. You keep the original hit but the beam cascade stops because the second hit was dodged.
If Twin-Linked Weapon:
1) Roll to hit with each die. "Whenever the ship suffers a hit, roll one die. If the die roll equals or exceeds the dodge score listed for the ship, all effects of the attack are ignored."
So if the target dodges, the hit did not happen. "Any Attack Dice for this weapon that do not successfully strike their target may be re-rolled," per the twin-link rule.
Based on the text of the rule it seems clear. Dodge is rolled right after the attack die is rolled (not at the end of the netire attack as is commonly done) and if the dodge is successful it cancels the effect of the hit.
Beams do not roll for extra damage because the original hit did not happen. Twin-link does re-roll dodged attack dice for the same reason, beause the original hit did not happen. The ship then gets to roll dodge again against the re-roll to see if it can dodge that one too.
Dodge with regard to CAF is a little less clear because the CAF rule says that the attack must "miss" and it is unclear that a dodged attack is a "miss". By strict reading, "all effects of the attack are ignored" is not the same thing as a "miss". Plain, common usage however, would indicate that "all effects of the attack are ignored", is the same as "miss."
Tzarevitch