부탁하다 means asking/requesting someone to give you something or to do something for you. it is a very common word (much more so than "request" in English) used for all kinds of requests in a variety of situations. 부탁 by itself is also used as a noun a lot to mean a request or personal favor (내 부탁 좀 들어 줘. 부탁이 있는데 들어줄 수 있어?). 1) 제 것도 부탁할게요 = I'd like to ask/request for mine too = Get me something/the same one too. You could say 제 것도 주세요 or 제 것도 해 주세요 (if it's an action) as well, but 부탁할게요 is less direct and more polite. 2) 다른 사람에게 부탁할게요. We say P-에게 N-을 부탁하다 (ask P for N) or P-에게 V-아/어 주기를 부탁하다 (ask P to V for you). So 다른 사람에게 means the speaker will ask someone else. 물어보다 is strictly for requesting information. Unlike "ask" in English, 묻다/물어보다 and 부탁하다 are distinct words that are never interchangeable. 3) 그럼 끓이는 김에 내 것도 부탁해. You could say 내 것도 만들어 줘 instead of 내 것도 부탁해, but when the thing being requested is already known, 부탁해 is both shorter and better. 부탁해 has the additional sense of "I'm relying on you" and even "Please!", so it's usually preferred over repeating what you're asking for.

When you want something to ask politely, any time you can use '부탁하다'. example 1. 제 커피도 부탁해요.(그 친구가 커피 사러 갈 때 커피를 부탁) 2. 제 것도 부탁해요(그 친구가 라면을 끓이려고 할 때 자신의 라면도 부탁) 3 소개 좀 부탁해요(누군가를 모를 때 옆의 사람에게 소개해 달라고 부탁) 5. 돈 좀 부탁해도 될까요?(돈을 빌리고 싶을 때 돈을 부탁) 6. 부탁해 아들! (아들에게 심부름 시킬 때 심부름을 부탁)

I think it is because 부탁하다 is asking for a favor etc.