Did Noah’s flood submerge the entire Earth, and was Noah sent to all of the Earth, or were Noah's people the only ones on the entire Earth?

This era is deeply ancient, and we have no certainty except what Allah has informed us of. You are correct in what you mentioned regarding Allah’s absolute justice, glorified be He, and indeed His perfect mercy towards His servants, and that He does not punish anyone except for their sin, nor does He hold anyone accountable except for what their hands have wrought. This is the refuge from every perplexing and problematic issue whose details are not clear to us: that we entrust its knowledge to Allah, glorified be His majesty, praised be He, the All-Knowing, the All-Aware, Who is not questioned concerning what He does, while they will be questioned.

It appears that Noah (peace be upon him) is the second father of humanity, and that the flood covered the earth and wiped out all human beings, except those whom Noah carried in the ark. The text in the Holy Quran states that the earth was then populated only by the descendants of Noah (peace be upon him), so all people on earth now are descendants of Noah (peace be upon him).

Allah Almighty says: (So We saved him and those with him in the laden ship. Then We drowned after that the remaining ones) [Al-Shu'ara: 119-120].

And Allah Almighty says: (And We made his progeny those who remained) [Al-Saffat: 77-81].

So, everyone on the face of the earth today, from all races of human beings, is descended from Noah's three sons: Shem, Ham, and Japheth.