Oh, dude, if you get me started on linguistics, we’ll be here forever! lol
Rahmah is the root word of both words used to describe Allah in the basmala: بِسْمِ اللهِ الرَّحْمٰنِ الرَّحِيْمِ, bismillahi al rahman al rahim, In the name of Allah the All-Merciful, the All-Compassionate. Al-Rahman is the quality of being entirely composed of Rahmah, so God is telling us that His essential nature is entirely Rahmah. Al Rahim is the quality of continuously showing Rahmah, so God is saying the He never fails or ceases to show Rahmah.
So, rahmah is often translated as “mercy” in English, but the context is more like mercy, caring, and love. The etymology of the word goes back to an ancient root that referred to the womb and actually if I remember rightly the Hebrew word for womb actually comes from the same Proto-Semitic root. So, at its basic level, rahmah refers to love and care on the level of a mother caring for and nurturing the child of her body. A mother has the power to neglect or harm her children or punish them harshly, but she doesn’t because she loves them and so she shows them mercy even in their failings. That’s the relationship of Allah with mankind and that’s why rahmah and derivatives are used more than any other word to describe God.
As for Yahweh, that’s another interesting linguistic question. Muslims consider the name of God to be Allah from Al-Ilah, “The God”. That shares the Semitic root “il” with Eloah, the singular of Elohim and Elohim has a really complicated relationship to YHWH, so it is sort of tangentially related, but Islam doesn’t include Yahweh as a name of God.