Well, I will take my best crack at explaining this as I understand it.
Adam was created. He was accountable to God for his actions. Eve was created from Adam, so not being of a "separate substance"; what ever "flaw" (or propensity to disobey) he possessed, she also possessed.
Adam and Eve existed in what I'd call a state of "probational righteousness". So long as they obeyed; the knowledge of good and evil did not become part of their awareness. Now this did not mean that good and evil did not exist in the world; they were just not aware of it. They were still capable of doing things that displeased God; but so long as they did not disobey a command given; they were not aware of any of what they may have done that was displeasing.
They had 3 commands:
1. Be fruitful and multiply
2. Keep and care for the garden. (Which is actually a military term meaning they were to protect it. Consequently they did not do a very good job at that because they did not expel the serpent from their presence.) Genesis speaks of the serpent being "cunning". Adam and Eve apparently recognized that; which I'll get into in a minute.
3. Don't eat of the tree of the knowledge of good and evil.
Now we know "evil" existed in the world before Adam and Eve sinned on account of two realities.
1. Satan fell before Adam did.
2. Genesis talks about "darkness" on the face of the deep. That had a physical dimension to it; but I believe it also had a spiritual dimension. And that aspect of the "origin of evil" I have a theory that this was a byproduct of God creating the universe.
Theory on the origin of evil
Now what about Adam and Eve (as well as the rest of the universe) made them (it) corruptible to begin with? Adam was created in God's image, but not with God's attributes. (omniscience, omnipotence, omnipresence, immortality, and eternally existent) God can not be corrupted by the knowledge of good and evil because of what He is. This explains both the fall of Adam and Eve; but why Jesus never sinned.
1 Timothy 2:14 tells us that Eve was deceived but Adam was not. Scripture says Eve ate the fruit because she wanted to be wise like unto God. Adam understood that eating the fruit was not going to end in the result Eve was looking for. So when Adam ate the fruit; he did so out of understood rebellion to God.
Note, nothing happened to Adam and Eve until Adam ate the fruit. This is where the concept of responsibility for sin / fallen nature passes from father to child comes from. Adam is the "buck stops here" of accountability of the two of them at the time. And this is because he was the first "of humanity" created. Adam and Eve together were "man created in God's image".
"Man" was created as gendered individuals because of the need for the capacity to reproduce. God knew creation would need that capacity because He knew the fall was coming and death would enter as a consequence of sin. God knew this because He knew "evil" existed from the point He'd created.
So this brings us to defining what is "original sin". One Eastern Orthodox poster defined it as the propensity to sin. I would define it as being on the receiving end of inheriting a fallen nature. That inheritance passes from father to child, not mother to child. "In Adam all die" 1 Corinthians 15:22. That's what I believe that passage means. All offspring of Adam inherited a fallen nature. This does not mean that the individual offspring have "sinned" yet; (Romans 9:11) but the inevitable outcome means all who are of Adam will sin.
So, to answer your question; does the sin nature pass from father to child? Yes.
Does it pass from mother to child? No.
Does the inheritance of the fallen nature have something to do with the "nuts and bolts" of reproduction. (The gametes of the material world.) No, it doesn't. This is how Jesus can be born of Mary, (literally created from her genetic stock) Mary being as sinner and Jesus not inheriting a sin nature from her. He was not of the "seed" of the first Adam.
Did a portion of the atonement "cover" the inherited fallen nature? Yes it did. It delivered the creation from that fall. Note, Adam and Eve (and the rest of humanity with them) were not the only ones affected by their sin. Their sin brought the "knowledge of good and evil" to all life in the universe via the fact that death now reigns. All life dies. And the atonement ultimately frees the creation from that consequence. This is how the new heavens and the new earth are created incorruptible (where there is no death). The atonement made that possible.
How is Jesus, living as an incarnated entity in His own creation; not be affected by the propensity to transgress. Why did Jesus not suffer the same inevitable fate Adam and Eve did? Because being directly created by the Holy Spirit as God nature inseparably joined to human nature made Him at least in the Divine aspect of what He was, incorruptible.
Now.... could Jesus sin? Yes He could. He as a human entity (just like Adam) had a will to decide what to do. Jesus though did not have the inherent disadvantage of being a temporal entity that is corruptible by the knowledge of good and evil. Which is what happened to Adam and Eve. They were corrupted by that knowledge. They were corrupted by that knowledge because they were not created with the eternal attributes of God. This is why they were made in God's "image" and not His "substance". Jesus was the incarnated "substance" of God.
Now because Jesus was the incarnated "substance" of God, is what prevented Him from sinning in the flesh. He was capable of living a life of perfect obedience because the "substance" of God was a permanent make up of who He/he was as an individual. He was the "God/man" which is what made Jesus inherently different from the first Adam.
Now what actually happened in Mary's womb? If you look carefully at the Greek. Gabriel tells her that the Holy Spirit will conceive of her "stock" the child. Biologically we understand that today as DNA.
A female egg contains all necessary chromosomes to create another organism even before its fertilized. This is how we have some organisms (like frogs for example) who can produce more frogs simply by laying eggs without a male fertilizing them. Birds do the same thing; they can lay unfertilized eggs. (Chicken eggs we eat are usually unfertilized.) Now all offspring produced of unfertilized eggs are always sterile females; yet not all unfertilized eggs (and kinds of eggs) create offspring.
A mammal egg contains everything a cell needs to be a cell. Sperm only contain DNA, protein (cell wall, flagellum - make them "swim", aspect that houses enzymes) enzymes that tell the female body the egg is fertilized, and a couple of mitochondria that propel the flagellum. Sperm aren't very complicated as far as cells go.
The human egg has all 46 chromosomes from the point of the development of the female's birth. When ovulation starts at the end of puberty, the ovary releases (usually) one egg at a time which still has 46 chromosomes, ready to be fertilized. If it isn't fertilized (or something has gone awry with the process) that egg is either reabsorbed or expelled from the body.
Now creation of sperm is similar process. Sperm aren't capable of being created though until after puberty. (They aren't present at birth, but the cells that will eventually produce them are.) That process of replication in males happens after puberty; where in females, it's part of the fetal gestation process. Where production of eggs though only creates two eggs; production of sperm creates 4 sperm. And unlike eggs, sperm only have half the chromosomes of what would be the normal compliment of the organism. So in the case of humans, sperm only have 23 chromosomes, not 46.
All eggs chromosomes are XX
Sperm are either X or Y.
Sperm determine gender of offspring. X=female; Y=male
So when a human egg is fertilized, it doesn't have 46 chromosomes, it has 69 chromosomes. And thus commences the process of which chromosomes are "kept" and which aren't. Once that "decision" is made; the extra 23 are discharged as a polar body.
Sometimes something goes awry in the process though and people are born with excessive genetic material. (Down's Syndrome is a whole extra (duplicate on the 15th) chromosome.) I don't think there is a human who's survived with less than 46 chromosomes. There aren't deletions of chromosomes, although there are deletions, or wrongful rearrangements of genes; that are the source of genetic disorders.
Genetics is a fascinating and amazing process; and genetic replication isn't just something that happens in the production of gametes, or at conception. Our individual genomes replicate and repair themselves every second of every day through the entirety of our lives.
So what happened with Jesus?
We know obviously He was born male. Born a male that was capable of reproducing. So what did the Holy Ghost do? Biologically; He basically took Mary's 46 chromosome egg and created a "male clone body" of Jesus's flesh. Now all the other things that are created as part of the person in the conception process (soul, body and spirit) were done of the Holy Ghost absent of a human contributor. (The part of what God contributes to our existence is the breath of life that makes us "living souls". All other aspects of us as human persons is part of the natural conception process.)
So, this difference of no human male contributor; made Jesus (the whole personhood of) a direct act of creation; (since there's no "place" biologically to get a Y from to make a male body). Jesus was not just a "Divine Soul" stuck into a human body. He was a whole person inseparably created of two natures: a Divine nature and a human nature. All that made up human nature was inherited from Mary (including human soul, spirit and flesh). All that made up the Divine nature was inherited from the Father; even though the male aspect of his humanity was a direct creation of the Spirit.
And here is where the concept of losing sperm or eggs being as being some "mortal sin" goes out the window. Sperm and eggs are lost all the time. They are either reabsorbed or expelled from the body and those process aren't necessarily connected to "intended sexual activity". Jesus too produced body fluids that "got outside" His body. Those processes are just part of biology.
I'm not sure if my explanation gave enough information to directly answer the question. "Original sin" or what I'd more accurately deem as a "fallen nature" is not specifically the presence of sin; because being born with this nature into a fallen world is not an act of the individual will, post Adam's choice.
And this is why I say it's a misnomer to say we have "free wills". We certainly have the ability to make choices that are independent of the wills and desires of other entities (God included) but that is not a will unencumbered by the fallen world and the fallen nature.
Jesus on the other hand truly had a free will because His decision making processes were not hampered by these other factors. He didn't have a fallen nature and He had the capacity to act to supersede the fallen world.
And here is where I believe was Jesus's biggest human temptation to sin; was to choose to act to supersede the fallen world outside of the permission He subjected Himself of the Father to perform. He stated that He only did what the Father directed Him to. Which makes Him the appropriate sacrifice because He is subject to governing rule of the Father, just as Adam was.
So there is my explanation for what ever I have clearly enough (and accurately) delineated to the reader that makes sense.