Rear derailer replacement.

Short of it. Wife picked up junk bike someone threw out, wants me to fix it for son-in-law.

Please note, I'm not assuming any quality here, she just wants to give him a bike to ride with his son.

It is a GT Agreessor Pro, Chinese made bike. It has a Shimano Acera derailer missing

the top small sprocket. The other one has 13 teeth.

I'm not knowledgeable about bikes but handy enough to do a replacement. If I had the sprocket and bolt,

I'd install that. Are those parts sold separately?

If I found another junk bike with a Shimano 13 T derailer any reason that wouldn't work.

If not, are derailers standard? Meaning, can I just order any derailer from Ebay or Amazon and it would fit?

Can I assume the GT Agreessor Chain is Standard?
EDIT to add,
This is a part I'm thinking about ordering, (if I need to order), I picked it because it has 13 T others don't.
Ah, need to give Ebay description, I don't have enough posts for to post URLs.
So here is an Ebay search term that brings up the derailer
SHIMANO RD-M360-SGS ACERA 7/8 SPEED BLACK DIRECT MOUNT REAR DERAILLEUR-NO PACKAG


Thanks for any enlightenment you can add, Mikek