Using non-metallic cable with this metal box having armored cable attached?
You can't use Romex. You need AC cable.

Your jurisdiction does not allow the use of Romex cable for electrical wiring. Romex is a brand name of a type of plastic cable called NM-B. 

I'm well aware that the Internet is full of instructables and videos that say to use Romex.  This is a cheaper wiring method that is not allowed in your town, for reasons.  

This is obvious from the builder's use of AC type armored cable and backstabs.  The AC type with steel boxes is a deluxe wiring method the builder would not have used if it wasn't compelled by law. 

If you have any doubts, go talk to your city's permit issuing office and ask them about wiring requirements there. 

But I only know Romex, I don't know anything about AC cable

It's not that big a deal. You use AC cable obviously. The cable clamps appear to be built in to the existing metal box. You'll need to make a bottom entry because all the top holes are full, then loop upward. 

So you'll need to get a metal box that has AC cable clamps built in.  The local electrical supply house can help.  If you bought from the local big-box, you can try there but their advice is terrible, so don't rely on clerks.  If you wonder why they're consistently terrible, it's because electricians and electrical supply houses constantly troll the big-box stores looking for competent clerks, and hire them away.