Learn Hardcore Korean: seonsaengnimege seomyeongeul badasseoyo. - I received a signature from the teacher.

With verbs of receiving like 받다, 에게 is understood as “from.” With verbs of giving like 주다, 에게 means “to.” You can also use 에게서/한테서 to make the “from” meaning explicit.

Should I use 선생님에게, 선생님한테, or 선생님께?

선생님께: most respectful (honorific for people of higher status). Very natural here.
선생님에게: neutral/standard, common in writing; fine in speech too.
선생님한테: casual/conversational.
All three are acceptable; choose based on formality.

What about 에게서, 한테서, or 으로부터 to say “from”?

에게서 / 한테서 explicitly mean “from” (에게서 is a bit more formal; 한테서 more colloquial).
으로부터 is formal/literary: 선생님으로부터 서명을 받았습니다 sounds very formal.
With honorifics you usually say 선생님께 (not “께서” for “from”—께서 is a subject marker).

Why is there an object marker on 서명 (서명을)? Can I drop it?

받다 is a transitive verb, so its object (서명) normally takes -을/를: 서명을 받다. In casual speech, Koreans often drop it: 서명 받았어요 is fine. In careful writing, keep the object marker.

Is 서명 the same as 사인?

서명: a formal/legal signature on documents, forms, contracts.
사인: commonly used for a celebrity’s autograph; many Koreans also say 사인 for everyday signatures, but in official contexts 서명 is preferred.
Note: the correct spelling is 사인, not “싸인.”

Can I change the word order?

Yes. 선생님에게 서명을 받았어요 and 서명을 선생님에게 받았어요 are both natural. Korean allows flexible order; place what you want to emphasize earlier.

How is 받았어요 pronounced?

It’s pronounced roughly as [바다써요] due to liaison and tensification. A practical romanization is badasseoyo.

What’s the politeness level of 받았어요, and how can I change it?

-았어요/-었어요 is polite informal (해요체).

Formal polite: 받았습니다.
Casual: 받았어.
Pick the level to match your relationship and context.

Can I say this more honorifically, focusing on the teacher’s action?

Yes: 선생님께서 서명해 주셨어요.
This highlights that the teacher did the signing for you, using honorific -시- and the benefactive 주다 → 주시다.

Is “선생님에서” ever correct here?

No. 에서 marks a place or source of an action/event, not a person as a giver/receiver. With people, use 에게/한테/께 (…에게서/한테서) or 으로부터.

Do I need to write “서명받다” without a space?

Write 서명을 받다 with a space (and usually the object marker). 받다 is an independent verb, so it does not attach to the noun like 서명하다 does.

Is the subject “I” omitted here? How would I include it?

Yes, it’s omitted by default. You can add it:

Topic: 저는 선생님께 서명을 받았어요.
Subject: 제가 선생님께 서명을 받았어요.
Use 저 (polite) rather than 나 in formal situations.

Could I say “배웠어요” or another verb instead of 받았어요?

Not for this meaning. 받다 collocates with 서명/사인 to mean “receive/get a signature.” If you want the act of signing itself, use 서명하다/서명해 주시다.

How can I show it took effort to get the signature?

Use a resultative like 받아 냈어요: 선생님께 서명을 받아 냈어요 implies you managed to secure the signature after some effort.