Maybe the course changed, but at this point, before "level 2", it did not teach continuous form. Continuous is introduced between level 2 and 3 as far as I can see... I got this sentence at first on a listening exercise. No way I could have got it right with only the things that have been taught until this unit.
I'll try, someone please correct me if I'm wrong.
I think the topic particle was omitted here because the sentence is informal/casual. But I think 너는 would have worked here.
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".
This is informal/casual speech. A more polite version could use "있나요?" or "있습니까?"
~나요 is just an ending to make something a question. You could use 있어요 too. This example just uses 있냐.
Check this website, it has casual question endings like 냐, 니: https://wiseinit.com/%EC%96%B4-%EC%95%84-%EB%8B%88-%EC%A7%80-%EB%83%90-%EB%82%98question-verb-endings-of-korean-korean-grammar-vs-grammar-2/
Polite question endings like 나요: https://wiseinit.com/polite-question-endings%EB%82%98%EC%9A%94-%EC%96%B4%EC%9A%94-%EC%95%84%EC%9A%94-%EC%A7%80%EC%9A%94-%EC%8A%B5%EB%8B%88%EA%B9%8C-korean-grammar-vs-grammar-3/
It is kind of a jump since we haven't been using informal speech or other question endings.
-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.
No, because there's no "something" in the original.
The phrase "are you talking to me?" is a famous line from the movie Taxi Driver, and in the movie, the character is being very rude when saying it. Most native English speakers will recognize this phrase because of the movie and recognize it as being very rude. The Korean is also very informal and would be rude in most cases.