It's common in informal American English (hat tip: @Damkerng) to omit the word "and" from the phrase "come and X". For example, the following pairs of sentences are equivalent:
Why don't you come and watch the movie with us?
Why don't you come watch the movie with us?
He can come and write his essay in our room.
He can come write his essay in our room.
Similarly, the phrase "come settle" in your sentence means the same thing as "come and settle".
Your sentences 1 and 2 are correct.
"Anyone can take anyone's help in my office" means that anyone in your office can take help from anyone else in your office. A more natural way of saying this might be "Anyone can accept anyone's help in my office" (meaning that people might offer their help, and it is ok to accept it from anyone), or "Anyone can ask for anyone's help in my office" (meaning that anybody can ask for help from anybody else).
"We have a friendly atmosphere in which anybody can approach anybody" is very naturally phrased, and means that any person in the office can approach any other person in the office.
In these (and maybe every?) cases, anybody and anyone are synonyms, so you could also say:
Anybody can ask for anyone's help in my office.
Or
We have a friendly atmosphere in which anybody can approach anyone.
I have seen arguments online that one or the other is more formal or polite, but I think this is subjective and not universally accepted, so it is safe to use whichever you prefer in formal or informal settings.
To answer your other question, to express that nobody knows the limits of any other person, you would say
No one in this office knows anybody's limits.
To say that nobody knows the limits of all their colleagues (for instance, I know the limits of my colleagues who sit next to me or whom I interact with often, but I might not know the limits of someone whose office is on the floor above me in a different department), you could say:
No one in this office knows everyone's limits.
EDIT:
Oops, Fluffy's comment is probably correct.
No one in this office knows their own limits.
means that no person in the office knows his/her own limits.
Best Answer
You have parsed this correctly: in the final sentence the verb and the ‘heavy’ conjunct subject are flipped so that the ‘lighter’ predicate is first and the ‘new information’ falls at the end of the sentence.