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Are there any sources that discuss whether Abraham and Noah knew of each other's existence, if there was ever any interactions between the two and/or if they ever met (not sure if their lives actually overlapped)?
Noah was after all Abraham's 9x-great grandfather.
The Kad Ha'Kemach by Rabeinu Bachyei ben Asher, seems to imply that Avraham saw Noach (see footnote ג). It is very possible he met him later on on life since they were from different locations. As was mentioned already in the comments Noach died when Avraham was 58 (siman נ"ח).
The Ramban 10:5 also explains that Avraham and his sons knew Noach and the testification of the flood.
Text of Kad Ha'Kemach:
Yes they Avraham stayed together with Noach for 39 years Midrash Sefer Hayashar parshas Noach states explicitely:
ויהי בצאת אברם מן המערה וילך לו אל נח ושם בנו וישב בביתם ללמוד את מוסר ה ואת דרכיו... ויהי אברם את נח שלשים ותשע שנים
When Avraham went out of hiding from the cave He went to Noah's house where Noach and Shem were, and he learrnt the ways of Hashem... and Avraham dwelled in Noachs house 39 years
Their lives did overlap. Abraham was born in 1948 (by the way, I love that fact given the events that happened in the secular 1948). Noah died in 2006. So Abraham was 58 years old on Noah’s death. An easy theory could be simply that most of us do not have a tradition knowing who our 9x great-grandfather is in eras where history was so much more easily recorded, and in between Noah and Abraham, the Tower of Babel incident occurred, so the connection was probably lost in those circumstances.
However, I found an unsourced different account on Chabad.org that I share below. I don’t know where this is taken from and would be grateful if someone on this website who may know could share that:
At the age of ten, Abraham decided to leave the cave and to go to old Noah and Shem, of whom his mother had told him many wonderful stories. Unknown to anybody, Abraham made his way from the low country to the mountainous region of Ararat in the land of Kedem, where Noah and his family lived. He was made welcome by old Noah and Shem, who taught him all they knew about G‑d and the ways of G‑d.
Abraham stayed there for nearly 39 years, until the year 1997. It was at the end of this period, when he was 48 years old, while still at Noah's house, that Abraham heard about the world-shattering event of the Tower of Babel, which took place in the land of Shinear, where Nimrod reigned supreme. Nimrod and his people wanted to build a tower that would reach up to heaven, so that they might establish their reign upon the heaven as well as on the earth. It was the height of arrogance and defiance of men against G‑d, and it led to confusion and to their dispersal and division into seventy tongues and nations. Abraham decided that it was high time for him to go out and teach them the truth about G‑d, and about the falsehood and worthlessness of the idols. He knew that in defying Nimrod, and even his own father, he would be risking his life, for Nimrod had proclaimed himself god and demanded that all the people worship him.
At the age of fifty (in the year 1998) Abraham returned to his father's house in Babylon
(https://www.chabad.org/library/article_cdo/aid/112063/jewish/Abrahams-Early-Life.htm)
The Midrash Shochar Tov (Tehillim #37) relates a fascinating dialogue between Avraham and Shem (Noach's son):
Avraham asked Malki-Zedek (=Shem): How did you [merit to] leave the ark?
Shem said to him: Through the [merit of] tzedaka that we did.
Avraham asked him: What tzedaka was there for you to do? Were there then poor people there? Only Noach and his children were there, so with whom did you do tzedaka?
Shem responded: To the wild animals, the beasts and the birds. We did not sleep the entire night, but rather we were feeding this one and that one. One time we were late [feeding the lion], and my father was left injured.
At that time, Avraham said [to himself]: These [people], had they not done "chessed" with the beasts, wild animals and birds, they would not have left from there, and because he delayed himself a little he received his "reward" and was injured. I, if I do "chessed" with humans who are in the image and likeness of the angels, then all the more so that I will be saved from harm. Immediately, "He planted an Eshel [in Beer-sheva]" (Bereishis 21:33) -- eating, drinking and לויה.
ואימתי קינא. כשאמר למלכי צדק, כיצד יצאתם מן התיבה. אמר לו, בצדקה שהיינו עושים. אמר לו, וכי מה צדקה היה לכם לעשות, וכי עניים היו שם, והלא לא היה אלא נח ובניו, ועל מי הייתם עושים צדקה. אמר לו, על החיה והבהמה והעוף. לא היינו ישנים כל הלילה, אלא היינו נותנין לפני זה ולפני זה. פעם אחת איחרנו את עצמנו, ויצא אבי משובר. אותה שעה אמר אברהם, מה אלו, אלולא שעשו צדקה עם בהמה חיה ועוף לא היו יוצאין משם, ובשביל שאיחר עצמו כמעט קיבל שכרו ונשבר, אני אם אעשה עם בני אדם שהם בדמות וצלם של המלאכים על אחת כמה וכמה שאנצל מן הפגעים. מיד (בראשית כא לג) ויטע אש"ל. א'כילה ש'תיה ל'ויה.
Noah and Shem were alive during the lifetime of Abraham
If we read the Bible literally, Genesis 9:28 says that Noah was alive for about 350 years. At some point in time, the flood occurs. This takes place in the year 1656. After the flood, Noah lives an additional amount of years before his death in 2006. Abraham was born in 1948 and was 58 years old when Noah died.
Genesis 11:10-11 says that Shem bore his first son at 100 years of age, just two years after the flood. Shem would go on to live for another 500 years, dying in the year 2258. So, Shem was around for 310 years after the birth of Abraham.
An alternative view
To put it in another way, it could be argued that biblical years were calculated differently until after the flood. This leaves you with two alternatives. Either the flood was local and Shem only lived an additionally 20-30 years after Abraham's birth or, the flood occurred long ago and Shem is not Melchizedek who says taught Abraham the Torah. Additionally, Rambam says Abraham discovered G-d on his own and cites a Midrash.
Conclusion
To put it differently, we simply do not know, begging the question.
