Question: Wouldn't 입니다 be spelled 임니다?
ip ni da ~ im ni da?
No, it is still 입니다. I asked my Korean teacher the same thing before. Whenever you see a pachim (받침, or last consanent in the syllable block like ㅂ in 입), the pronunciation depends on the next syllable. You do not pronounce the ㅂ when it is followed by a consanent (ㄴ in 니). Instead, you can think of pressing your lips together as if you're going to make the ㅂ sound. However, you do not complete the ㅂ sound since it is followed by ㄴ. So yeah, it does sound like 임 when you speak it, but it is spelled like 입.
같이 sounds like 가치
For more info: http://www.koreanwikiproject.com/wiki/index.php?title=받침