Many questions on this site (usually about Catholic/Marian doctrine) have had references to sin as being inherited/transmitted from the father (and specifically the father, as opposed to either/both parents), almost as if it were some kind of genetic disorder, which is an idea I've never encountered before. Is this official Catholic doctrine? Are there many/any Protestant denominations that include a similar belief?
The magisterial text that is closest to the subject in the question is canon 2 of the "Decree on original sin" of the ecumenical council of Trent :
If any one asserts, that the prevarication of Adam injured himself alone, and not his posterity; and that he lost for himself alone, and not for us also, the holiness and justice, received of God, which he lost; or that he, defiled by the sin of disobedience, has only transfused death, and pains of the body, into the whole human race, but not sin also, which is the death of the soul, let him be anathema; inasmuch as he contradicts the apostle, who says: By one man sin entered into the world, and by sin death, and so death passed upon all men, in whom all have sinned.
Note that the canon does not mention Eve. Interpreting this text literally, it can be inferred that the only sin that mattered regarding its propagation to the future offspring of Adam and Eve was that of Adam. Therefore, the answer to the first part of the question is affirmative: original sin is propagated by the father irrespective of the state of the mother.
This is fully consistent with the need for the Incarnated Word to be conceived virginally: if He was not a patrilineal descendant of Adam, He would not inherit original sin even potentially and therefore would not need to preventively redeem Himself in view of his own future merits (as opposed to the case of St. Mary, who was preventively redeemed, i.e. preserved from original sin, in view of the future merits of Jesus).
St. Thomas Aquinas dedicated an article of his Summa Theologica to this very issue :
Article 5. Whether if Eve, and not Adam, had sinned, their children would have contracted original sin?
I answer that, The solution of this question is made clear by what has been said. For it has been stated (Article 1) that original sin is transmitted by the first parent in so far as he is the mover in the begetting of his children: wherefore it has been said (Article 4) that if anyone were begotten materially only, of human flesh, they would not contract original sin. Now it is evident that in the opinion of philosophers, the active principle of generation is from the father, while the mother provides the matter. Therefore original sin, is contracted, not from the mother, but from the father: so that, accordingly, if Eve, and not Adam, had sinned, their children would not contract original sin: whereas, if Adam, and not Eve, had sinned, they would contract it.
Regarding the second part of the question, it does not follow from the canon that what is propagated is "some kind of genetic disorder". First and foremost, the "death of the soul" is the deprivation of sanctifying grace and charity, which are spiritual gifts. Second, as far as the canon is concerned, it is legitimate to interpret that the exposure of Adam and his offspring to "death, and pains of the body" was not due to some genetic mutation after sin but to the cessation of the divine intervention that shielded Adam and Eve, before they sinned, from external attacks (from lightning, predators, germs, etc.) and internal decay. Which is clearly the only interpretation that is compatible with contemporary scientific knowledge.
 Thomas Aquinas, ST I-II, Q. 81, Art. 5. https://www.newadvent.org/summa/2081.htm#article5
Alternative translation by Alfred J. Freddoso: https://www3.nd.edu/~afreddos/summa-translation/Part%201-2/st1-2-ques81.pdf