Equivalent of: "Ya talkin' ta me?!"

This is more aggressive when you leave off the "are"

-에게 is a suffix that you attach to nouns to imply "TO" or direction. -에 alone can be used though that's more for places in my personal experience, and you almost always use 에게 with people. To give an example: 나는 집에 가요 - I go (to) home Sarah에게 편지를 보내요 - I send a letter TO Sarah.

If you want to say from, 에게서 sooooo Sarah는 저에게서 편지를 바다요. - Sarah receives a letter FROM me.

Adding 에게 means that an action is directed towards something/someone. So writing 나에게 means an action happened "to me."

I believe 있냐 is a very informal version of 있습니까 - it implies a question, but it is extremely casual/could be taken as rude.

너 미쳤냐?*

오 그래!

When you're friends with that person (same year born) or really close to them

Yes, in this case it is being rude on purpose for extra aggression and insult.

I'll try, someone please correct me if I'm wrong.

1. I think the topic particle was omitted here because the sentence is informal/casual. But I think 너는 would have worked here.

2. That "고" is not "and". The structure used there is verb stem + 고 있다, which is the present progressive tense. In the sentence above the verb is 말하다, the verb stem is 말하, then you add ~고 있다. So the end result is 말하고 있어요 which is "talking".

3. This is informal/casual speech. A more polite version could use "있나요?" or "있습니까?"

"Are you speaking with me?" is also incorrect

I'm thinking it's "나에게 말하고 싶은 것이 있나요?" But I'm not sure if that sounds natural.

Pedagogy (or 교수법), indeed!!

N.B. '교수'= give teaching 법= method