Adam is the father of all humanity. He was literally the first father, and the measure of how good a father he was has repurcussions for literally every human on earth.
Looking for any sources that discuss anything related to Adam specifically as a father (Chava is explicitly called the mother of all life, so I am not asking about her yet).
The Ohr HaChaim Hakadosh makes reference to this, albeit just as an aside, when speaking about how Chava was formed from Adam's side.
On his commentary on Bamidbar 21:7 he writes:
והכוונה היא על זה הדרך הגיא אשר בא מפאת האשה שהיא חוה שבאה מאב כל הנבראים שלקחה ה' מצלעותיו ואין אשה בעולם שבאה כדרך זה מצלעות האדם כי אם זאת
The meaning of the verse then would be as follows: "the valley (grave) which was a state reached due to woman, who had been formed out of the "father" of all mankind, Adam, is the only one who had been created in this fashion, part of the rib of Adam, her husband." (Sefaria transaltion).
This choice of expression is also adopted by Rabbeinu Bachya. The verse in Bereishis 19:32 when discussing Lot and his daughters says:
לְכָ֨ה נַשְׁקֶ֧ה אֶת־אָבִ֛ינוּ יַ֖יִן וְנִשְׁכְּבָ֣ה עִמּ֑וֹ וּנְחַיֶּ֥ה מֵאָבִ֖ינוּ זָֽרַע׃
Come, let us make our father drink wine, and let us lie with him, that we may maintain life through our father.”
Rabbeinu Bachya there notes:
והנה חטא האם וחטא הבנות במעשה הזה שוה ע"י היין שהוא הגורם רעות העולם, ואולי רבוי את אבינו, את אבינו הראשון, שגם הוא שתה ע"י אשה
In this instance the sin committed by the mother (Chavah) and the daughters (Lot’s) was identical in that it was committed by means of wine which has brought much tragedy on mankind. Perhaps, the additional word אבינו, “our father,” in this verse is even an allusion to our original/first father (i.e. the father of mankind, Adam). It reminds us that he too had been given wine to drink by his wife.
Probably, it would be wrong to think of Adam (or Noah) as a Jewish father, because only regarding Abraham (Bereshis 18:19) it is stated that he was instructing members of his household. Possibly, prior to Abraham's times it was not necessary, because people in earlier generations could easier achieve their perfection. However, one could look at Adam as a the original source of inspiration and education based on who he was and mistakes he made, just like we look at generation of Midbar. They do not have portion in Olam haBa but we learn from their experience, from their achievements and failures. For example, Adam was the first parent who learned something important from his son (the power of teshuva), Bereshis Rabba 22:13.
