Did Jesus teach that marriage existed before the Fall?
In Matthew 19, Jesus uses the Jewish significance of scripture to answer a question about divorce in a way that challenges the Jewish significance of scripture.
He answered, “Have you not read that he who created them from the
beginning made them male and female, and said, ‘Therefore a man shall
leave his father and his mother and hold fast to his wife, and the two
shall become one flesh’? So they are no longer two but one flesh. What
therefore God has joined together, let not man separate.” Matthew 19:
4-6
Jesus referencing scripture is not necessarily Jesus verifying that what is written/read is truth - only that it is written/read. The Sermon on the Mount is a perfect example of this.
What Jesus teaches from this particular reference to Genesis is not that marriage existed before the Fall, but that, once married, a man and his wife 'are no longer two but one flesh.' When the law of Moses allowing divorce is brought up, Jesus dismisses this law as a concession to the Jewish people's 'hardness of heart', and reiterates his teaching:
"And I say to you: whoever divorces his wife, except for sexual
immorality, and marries another, commits adultery.” Matthew 19:9
Does Jesus suggest that marriage and death go together?
Jesus is responding in Luke 20 to 'the Sadducees, those who deny that there is a resurrection', who are trying to show the incompatibility of scripture with the resurrection by their hypothetical wife of seven brothers. The reason for the law of Moses referenced by the Sadducees is revealed in the original scripture:
If brothers are living together and one of them dies without a son,
his widow must not marry outside the family. Her husband’s brother
shall take her and marry her and fulfill the duty of a brother-in-law
to her. The first son she bears shall carry on the name of the dead
brother so that his name will not be blotted out from Israel.
However, if a man does not want to marry his brother’s wife, she shall
go to the elders at the town gate and say, “My husband’s brother
refuses to carry on his brother’s name in Israel. He will not fulfill
the duty of a brother-in-law to me.” Deuteronomy 25: 5-7
Marriage in both Moses' and Jesus' time was deemed a solution to the threat of non-existence in death. It provided a way for men to ensure that children born to a particular woman continued both their genetic code and their name - evidence of their existence - after their death. So marriage is a way of avoiding this sense of non-existence that comes with death, by ensuring one's name (self) lives on.
The necessity of carrying on a name through children within a marriage is only required when one faces death as the end of one's existence. But this is not the case in the resurrection:
And Jesus said to them, “The sons (and daughters) of this age marry
and are given in marriage, but those who are considered worthy to
attain to that age and to the resurrection from the dead neither marry
nor are given in marriage, for they cannot die anymore, because they
are equal to angels and are sons (and daughters) of God, being sons
(and daughters) of the resurrection." Luke 20: 34-36
If they cannot die anymore, then they need not strive to replicate themselves in order to carry on their name. Therefore, they need not marry nor have children, because their name continues through living to God. Jesus then went on to demonstrate how Moses understood that Abraham, Isaac and Jacob all continue to 'live to God' in this way, and their name is carried on:
But that the dead are raised, even Moses showed, in the passage about
the bush, where he calls the Lord the God of Abraham and the God of
Isaac and the God of Jacob. Now he is not God of the dead, but of the
living, for all live to him.” Luke 20: 37-38
So, is it better not to marry?
This is also linked to the following passage in Matthew, when the disciples suggest that 'it is better not to marry':
The disciples said to him, “If such is the case of a man with his
wife, it is better not to marry.” 11 But he said to them, “Not
everyone can receive this saying, but only those to whom it is given.
12 For there are eunuchs who have been so from birth, and there are
eunuchs who have been made eunuchs by men, and there are eunuchs who
have made themselves eunuchs for the sake of the kingdom of heaven.
Let the one who is able to receive this receive it.” Matthew 19: 3-12
Jesus suggests here that there are those who choose not to marry nor have children 'for the sake of the kingdom of heaven'. I would imagine that Jesus counts himself as one of these, as would Paul and many later saints. The kingdom of heaven is then built through their example: showing that they need not have offspring or marry in order to carry on their name. Like Abraham and Moses and Jesus, their name continues in the way they live to God.
How does one live eternally by living to God?
The way it's described, this way of carrying on one's existence sounds a lot like fame. But building monuments, inscribing one's name on everything, getting one's name up in lights or their face on TV - these are not eternal.
Elvis 'lives' for his fans - not through his fame, his music or his movies, but through the continuing impact on their own lives of the unique way he connected with the world. Jesus lives for believers in this same way - not through the written word or documented proof, but because the way that his unique expression of God's love and connection impacted on the lives of writers in the early church continues to impact on the lives of believers today.
Jesus' name was never documented during his life, nor was it carried on through marriage or offspring - in this way, he experienced not only death, but the apparent non-existence that comes from death without his name being carried on. His name was essentially 'blotted out from Israel'.
Yet the name of Jesus lives on over two thousand years later in the way that he connected with others: his examples of unconditional love, fellowship, non-judgement and inclusion, in what he shared of his relationship with God, and his invitation for others to connect with God in the same way, via his living example.
Conclusion
We married initially because we feared that death brought about non-existence: we are driven to replicate ourselves through offspring guaranteed to be ours, and carry on our name (an extension of ourselves) beyond our death.
The resurrection obviates the need for marriage and children for those who are considered worthy, by ensuring one's name, their existence, continues beyond death as they 'live to God': ie. express God's love and connection through the unique way that they connect with the world.