Essential elements or not?

Hi. Would you say what follows (follow??) the comma and what are underlined are essential although they have commas before them and somewhat away from the nouns/words they modify, "a girl" and "Jane Doe's puppy"? (I think they modify those words - not sure, though) Also, would you write the sentences differently?

He saw a girl, who lives next door to him, throwing a piece of paper at a boy who was sitting next to her.

At that moment, he looked like Jane Doe's puppy, which has (had?) a big appetite, throwing a fit if it didn't get its meal on time.

As I think the next sentence is somewhat related to the topic, I wish you would tell me if the underlined is an non-essential element or not. I think it is non-essential.

He is like Jane's puppy, which has (had?) a big appetite, yelping nonstop until it gets its meal when when the meal doesn't come on time.