Are there any commentaries which shows that the daughter of Pharao refuted idolatry, made a act of repentance or anything like that? I wonder what it was that made her save Moshe, that she had compassion for him, while she was raised by a father that probably told her Hebrews were no good.
According to the Gemorah (Megillah 13a), she was actually going to the river with the purpose of converting to Judaism:
רבי שמעון בן פזי כי הוה פתח בדברי הימים אמר הכי כל דבריך אחד הם ואנו יודעין לדורשן (דברי הימים א ד, יח) ואשתו היהודיה ילדה את ירד אבי גדור ואת חבר אבי שוכו ואת יקותיאל אבי זנוח ואלה בני בתיה בת פרעה אשר לקח מרד אמאי קרי לה יהודיה על שום שכפרה בע"ז דכתיב (שמות ב, ה) ותרד בת פרעה לרחוץ על היאור ואמר רבי יוחנן שירדה לרחוץ מגילולי בית אביה
R. Shimon b. Pazzi when he would begin an exposition of the Book of Chronicles would say the following: “All Your words are one”, and we know how to find their inner meaning. [It is written], “And his wife the Jewess bore Yered the father of Gedor, and Hever the father of Socho, and Yekutiel the father of Zanoah, and these are the sons of Bitya the daughter of Pharaoh, whom Mered took” (I Chronicles 4:18). Why was she [the daughter of Pharaoh] called a Jewess? Because she repudiated idolatry, as it is written, “And the daughter of Pharaoh went down to bathe in the river,” (Exodus 2:5) and R. Yohanan, [commenting on this,] said that ... she went down to cleanse herself from the idols of her father’s house.
I hope this helps the second part of your question. This is a direct quote from "The Torah treasury" from Artscroll (p. 139)
The answer lies in the order of what transpired. First Pharaoh's daughter took pity on the child. Realizing that his cry aroused in her an unnatural compassion, one that ran contrary to her home and upbringing, she concluded that it must be a reflection of the natural sense of mercy that is the heritage of the Jews. The crying child aroused a spark of mercy inside her that came to fruition when she converted to Judaism. (R Aharon of Sanz)