The correct way to say the first sentence is: 1. 안녕하세요! 저는 앰버예요. 그리고 아미는 미국서 왔어요 (note 저 and 왔). 2. 안녕하세요! 우리는 앰버와 아미예요. 미국에 살아요 = Hello, we're Amber and Army from America. #1 sounds like something you'd say in person with your friend next to you. It's odd in a letter, although possible. #2 might be better if you're writing an informal letter or email directly from the USA. The phrases are already in polite form. You should never say 앰버십니다 because 십 makes it respectful to yourself. 앰버입니다 is fine. It's just more formal sounding than 앰버예요. And 팬레터 is the more commonly used term for "fan letter". 팬(의) 편지 is also possible but not used much.

안녕하세요 ! 앰버씨 방탄서녀단 팬이시죠 ? The A.R.M.Y here means that you are a fan of BTS, am I right ? I am a Korean learner as well. I believe you can say. 안녕하세요. 저는 미국에서 아미, 앰버라고합니다. Lets hear more answer from natives !