People really don't use hanja, so you can get really far by just differentiating them based on context. Unless you want to go into an academic study of Korean, you really shouldn't need to learn it.
However, if you're also considering learning Japanese or Chinese, it may be worthwhile since there's a lot of overlap.
돌 can mean either "first birthday" or "rock, stone", it's just two words of different origin that happen to look the same in modern language (like English "sea" and "see" happen to have the same sound). It was probably the "first birthday" meaning that caused it to be included in this lesson and then somebody rightly complained that the translation "rock" is also correct. I don't know why you got corrected though; "first birthday" was accepted for me.