Good question! 1. In grammar (1) Propositive sentence / Imperative sentence - (으)러 can be used with '(으)ㅂ시다 (Let's) and '(으)세요(Please V~)' (Ex: 밥 먹으러 갑시다 (Correct) - But '(으)려(고)' cannot be with those '(으)ㅂ시다' and '(으)세요'. (Ex: 밥 먹으려고 갑시다 (Wrong) (2) Tense - (으)러 can be used for the present, past, and future tense. - But (으)려(고) cannot be used for the future tense. (Ex: 밥 먹으려고 나갈 거예요 (Awkward) 2. Nuance 1) 친구 만나러 까페에 가요 2) 친구 만나려고 까페에 가요 Both are okay in grammar. But in terms of nuances, in my personal opinion, "1) 친구 만나러 까페에 가요" is relatively a little more focused on the 'going to cafe' part, while "2) 친구 만나려고 까페에 가요" is a little more focused on the purpose part "to meet (my) friend". If I should say "I'm going to cafe to meet (my) friend" in Korean, I would not use "만나려고", and I would definitely use "만나러". The only time that I may use " 만나려고" would be to explain the purpose to go out, or to answer to the question "Why are you going to cafe?" Hope this can help you!

It sounds good as it is, but it may be better if you change some parts a little bit as follows: 제가 한국어를 (말)할 때 틀린 부분이 있으면, 이야기 (or 말씀) 해 주시면 감사하겠습니다 - I thought it would be better to put '제가' at the beginning - 틀린 부분이 있으면: If there are 'some parts that are wrong' - It may not be your whole sentences but just some parts, which is corresponding to 'a mistake' in English. - As we already put '제가' at the beginning of the sentence, we can drop '저한테 (to me)'. Korean likes to drop the objects if it is clear enough without it. - I used 이야기 or 말씀 with '해 주시면' instead of just 말하면 to sound more formal or polite. Hope it can help you!

This may help you! https://youtu.be/cpro961BELY?si=36gi0jqvi9mfE8w9

Accidentally can be "실수로" in Korean, but 실수로 actually means "by mistake" in English. "By mistake" and "Accidentally" may be slightly different in English, right? So, alternatively, if your intention is "accidentally" not "by mistake" for speaking not for writing, you can say "모르고" which means like "unconsciously" or "without knowing". Another tip, if you end your verb with 아/어/해 버렸다 (버렸다 is the past form, and 버리다 is the present form), it emphasizes that it just happened or became without your intention. I pressed it on accident/accidentally = 나 모르고 (or 실수로) 그거 눌렀어 (눌러버렸어 is better) I accidentally knocked the cup over = 나 모르고 (or 실수로) 그 컵을 쏟았어 (쏟아 버렸어 is better) I accidentally told the secret = 나 모르고 그 비밀을 말했어 (말해 버렸어 is better) For accent, I agree with 억양 or 말투.

"는다/다"는 Formal Plain Ending 이에요. - Formality에는 Formal과 Informal이 있고, - Honorific Level에는 Honorific과 Plain이 있어요. 논문은 교수님을 위한것이 아니라, 불특정다수 (unspecified individuals)에게 공개하는 공식 문건이에요. 교수님은 단지 그 문건이 공개되기전에 Review만 하는거지, 논문이 교수님을 "위해"작성하는 것은 아니에요. 즉, 교수님 = not a reader, but a reviewer 논문의 독자 (reader) = 불특정다수 (unspecified individuals) 논문의 독자가 불특정다수 (Unspecified individuals)이기 때문에 독자의 Honorific Level을 정의할 수 없고, 이럴땐 Plain Ending을 사용해요. 그리고 논문은 공식문건이기 때문에 Formality만 반영해요. 따라서 논문에는 Formal Plain Ending을 사용해요. 이렇게 독자를 특정할 수 없는 신문기사나 기타 공식문건들에도 다 Formal Plain Ending을 사용해요.

"It was raining so I brought umbrella" In English, the past tense should be shown for both "was" and "brought". But Korean is less strict in grammar, so the tense can be shown one time at the last verb ending only. So 비가 오길래 우산을 가져왔어 is the standard sentence with the past tense at the end, 왔어. So, if you use 비가 왔길래, then it is for emphasizing "was raining". And emphasizing is additional not mandatory depending on speaker's intention. This goes the same for other conjugations at the middle such as: 비가 와서 우선을 가져왔어 -> Standard 비가 왔어서 우산을 가져왔어 -> Additional for emphasizing 비가 오기 때문에 우선을 가져왔어 비가 왔기 때문에 우산을 가져왔어 비가 오니까 우선을 가져왔어 비가 왔으니까 우산을 가져왔어 Etc.