Don't know if I entirely agree either that you're "born with it or not", but I think it is something largely unteachable, but most people do develop it over time (if they're going to develop it at all). Hence, teenagers often don't have much but they will get it eventually, or certain life experiences can open you to having more empathy.
Sure you can. It's like any life skill. There might not be a systematic universal way to learn "professionalism", "manners", "etiquette", or "communication skills", but those are certainly taught and learned. Why not empathy, which is absolutely a core part of each of the above?
It's a brain function to have instantaneous access. You can't teach the brain to mylenate faster. However, teens can slow down (count to 10) and then they can access much of the same as adult judgement and empathy.
There isn't much you can do to "teach" it.