That's correct. Like it's other connective counterpart, 어서/아서, it both can provide reason for something and conjoin two clauses in sequence. In (으)니까's case, however, this sequential conjoining can be translated as "When (I) [first clause], I discovered/realized that [second clause]".
Thus, for your sentence
집에 도착하니까 비가 내리기 시작했다
a translation would be "...
The way I see it, 지 when used without a pronoun like '어디'(where) generally stands for an unknown or uncertain thing, whereas 것 usually stands for something that is known. (걸 is just a contraction of 것을).
Taking your second sentence,
아이들은 아기가 자는 걸 몰라요 => The children don't know that the baby is sleeping.
Here, the fact that the baby is actually sleeping ...
blimpy and Rathony's answers are useful, so I will answer why other options are incorrect.
-거나: "A하거나 B하다" implies that A and B are both possible options. However, selecting "집에 도착하다" or "비가 내리기 시작하다" is unnatural.
-거든: If you write "집에 도착하거든", then it implies "If you arrive home, then ..." However, there is no causality between arriving home and raining.