As helpfully pointed out by Grammarly, a comma before and should only be used if it's linking two independent clauses. The clauses here are
We looked at the city
found that beneath every surface
a whole world of adventure was waiting for us.
You're quite right that the first comma is misplaced, but it's because "found that beneath every surface" is not an independent clause, not because it's splitting up the "we" and the "found". If it had been "We looked at the city, and we found that..." the comma would have been fine.
As for your question about the grammatical name for "beneath every surface" I'm not sure there is a specific term for it. It's just a clause with an adverb, beneath, modifying the noun phrase every surface. The context should tell the reader that this is referring to every surface present in the city "we" looked at, as this was the last object mentioned in the sentence.
I would say (though this is mainly opinion without anything to back it up so it may not really belong here) that even the second comma is optional. The sentence would be just fine as
We looked at the city and found that beneath every surface a whole world of adventure was waiting for us.
This does run the risk of becoming a run-on sentence though, and the comma does help to separate things in a way that's easier to parse.