In English, it's not incorrect to have two consecutive repeated words in a sentence, and one place you'll encounter that is a phrasal verb followed by a preposition.
Nevertheless, other shows went on on the fatal day.
(H.L. Mencken, On Politics)
They scuttled the vessel off the harbour's mouth, and came in in the boat.
The phrase is a "take-off" Definition: (1) (2) or copy of a famous phrase:
Down the Rabbit Hole
--- Lewis Carroll, Alice's Adventures in Wonderland (also called Alice in Wonderland) (3) (4)
To "head down the rabbit hole," or "head down a rat's hole", simply means to go into it and downwards. (See definition)
The meaning of "go ...
It's a bit of a funny one as I believe it's considered colloquial rather than "proper" English, but to head in [a direction] just means to go towards that direction.
A similar question has actually been asked in the English Language & Usage SE.
It's also worth noting that you can use it to mean a literal direction:
Headed down the street. (...
That's a common phrase in football. The meaning might not be obvious at first, but once you understand it it's actually quite a literal description. It describes a type of pass typically played towards a team mate who is already running.
Instead of being passed directly towards the player's feet, the ball is passed into empty space ahead of the player, at ...
I'd say you were right on with your interpretation "move to the rat's hole": it is a common intro for video game/movie descriptions, to use a flowery version of:
Enter the world of [theme]
In this case, just a clever way to introduce the idea that you're going to be playing as a rat (enter the world of rats).
Along the lines of you could start a ...
According to Macmillan Dictionary, work out means:
"to solve a problem by doing a calculation"
I was born in 1947: you work
out my age.
So yes, the sentence means that the boy solves (finds the answers to) his questions earlier, in case he gets lazy in the future.
"Getting along" does not fit. In fact, even your example is wrong. It should be "I wonder how Michael is getting on in his new job", although some people do seem to mix up these two idioms. "Getting along" usually refers to a relationship between two people.
"Compromise" means a trade-off between two things and can sometimes be an agreement to settle ...
"Go around doing" is a completely neutral expression. It is neither good nor bad. You have to determine that from context.
For example, you could write your sentence 1 as the positive:
My daughter goes around telling everyone how much she loves me.
However, "keeps going around" is slightly pejorative, because the use of "keep" implies doing ...
I can think of two meanings of "stand with".
To be/remain loyal to someone (Which is your case)
To physically remain with someone
She stood with me around the campfire.