안방 typically refers to the master bedroom or the parent's room. I would have said 아기의 방에 들어가요
what if they gave the master bedroom to the baby
If they gave the master bedroom to the baby then parents' bedroom is referred to 안방. The meaning of 안방 is 안주인이 거처하는 방 = the lady of the house's room.
Living in Korea I've mostly heard 안방 as bedroom. Is there a reason 침실 is used here?
It is 寝室 as in Chinese. I am not a Korean, so best ask others.
The frog enters the baby's bedroom
Get outta there, frog
Does 들어가요 mean "to go into/enter" here?
Yes it does :)
The baby : DASHI RUN RUN RUN !!!