If a boy was born in a foreign country but was immediately adopted by a US citizen and immigrated to the United States while holding a temporary immigrant visa, once the visa has expired (10 years residency in the US) can the boy apply for US citizenship? If yes, how soon?

You can't get citizenship without a valid green card. Get that first and then he's on his way. Shouldn't be a big deal if his Temp immigrant visa hasn't expired.

he is probably a US citizen, based on derivative citizenship.

It depends on the exact facts of his adoption, the IV, entry to the US with as a PR, his parents citizenship and his parents timeline for physical presence in the US. He likely derived US citizenship from the day of entry as a permanent resident.

He has a 10-year PR card? Please confirm this. Get this information and consult with an immigration attorney that us experienced in naturalization and citizenship. Many attorneys will not be familiar with derivation and acquisition. Ask them specifically if they are familiar with these two terms.