‘get into’ here suggests that it isn’t something easy to do. There is a problem to solve. The cleaning lady, on the other hand, can simply ‘go into’ each room. It’s her job.

Yes :)

Go into is simply to go. Get into has an inference of breaking and entering rather than straight entry. Or perhaps you need a special invite or special pass to get in somewhere. Whatever the means, it's not just a matter of simply walking in.

Both mean “to enter in”. In the second story it sounds like the door is locked and so the speaker needs to find a sneaky way to break in, to get in, maybe by picking the lock.

Get into is correct in the first part, in this case to get into is in the present tense. The story/memory/recollection of events is written in the present tense. Up until that point when discussing getting into his room. Go into is also the present tense, and would be referring to going into his room in the present right now as it was happening. . It is possible to say "go into his room" for the first part also. it would not change the meaning. Get into is possibly implying going entry illegal or without permission. Get is very common in English we use it all the time for many situations likewise with the word got.