1. If Christ's death is the cause of life eternal, and Judas is the cause of Christ's death, then wouldn't that imply that Judas cause is life eternal?
No. To quote St. John Chrysostom's anaphora, "On the night when He was betrayed, or rather when He gave Himself up for the life of the world..." (HC translation). Whether Judas betrays Him or not, it is only with His will, decision, and direction that the Passion takes place. He first had to voluntarily submit Himself to the events to come before they could take place - otherwise, as the Devil points out when tempting Christ, 12 Legions of Angels could have easily shown up.
2. If Judas (or anyone) hadn't betrayed Jesus and Jesus didn't die -- would we have been saved?
Yes. We clearly state that the Lord knew from even before Adam's fall that He would save us, and that when man fell, it was a forgone conclusion that He would do something to redeem us... To quote St. Basil's anaphora:
But when he disobeyed You, the true God who had created him, and was led astray by the deception of the serpent becoming subject to death through his own transgressions, You, O God, in Your righteous judgment, expelled him from paradise into this world, returning him to the earth from which he was taken, yet providing for him the salvation of regeneration in Your Christ. For You did not forever reject Your creature whom You made, O Good One, nor did You forget the work of Your hands, but because of Your tender compassion, You visited him in various ways: You sent forth prophets; You performed mighty works by Your saints who in every generation have pleased You. You spoke to us by the mouth of Your servants the prophets, announcing to us the salvation which was to come; You gave us the law to help us; You appointed angels as guardians. And when the fullness of time had come, You spoke to us through Your Son Himself, through whom You created the ages. He, being the splendor of Your glory and the image of Your being, upholding all things by the word of His power, thought it not robbery to be equal with You, God and Father. But, being God before all ages, He appeared on earth and lived with humankind. Becoming incarnate from a holy Virgin, He emptied Himself, taking the form of a servant, conforming to the body of our lowliness, that He might change us in the likeness of the image of His glory. For, since through man sin came into the world and through sin death, it pleased Your only begotten Son, who is in Your bosom, God and Father, born of a woman, the holy Theotokos and ever virgin Mary; born under the law, to condemn sin in His flesh, so that those who died in Adam may be brought to life in Him, Your Christ.