Could "go" here also refer to things that would in English be translated as "it", such as when it refers to a dog (since it's masculine in Polish)? Thank you!
Yes, it could indeed. As long as it's masculine in Polish.
With Genitive, it could be either masculine or neuter.
none in meaning. "Jego" is used in accented position in the sentence (always at the beginning, often at the end if a sentence is longer) , or to stress the word, "go" is used only in not accented position in a sentence.
Accented position is also after preposition, but genitive=accusative form of "on" has "niego" for that.
You don't necessarily important to have the "We", as it just means "Got him". However in an exam they may not be that kind