Q:

Neither L nor S lives in either A or B, What is your inference regarding this sentence? 

Can "Neither Nor" and "Either Or" appear in the same sentence?  If yes, what is your inference regarding the sentence below? 

"Neither L nor S lives in either A or B"

The answer my friend thinks is "No one (i.e L,S) lives in any place that has been mentioned (i.e. A,B)", But for me, some logic seems to be missing. 
I believe the sentence can be written like this. 

"Neither L nor S lives in A"

or

"Neither L nor S lives in B"

What I think is "One (L or S) cannot live in the places that have been mentioned  (i.e. A or B) So the other one can live in A or B"
i.e. "If S cannot live in A or B, L lives in A or B" or "If L cannot live in A or B, S lives in A or B". 
Can anyone help find the correct answer? 

A:

Let's start with pointing out that mathematics and formal logic are two things that do not apply to natural language in the sense that one can not follow clear rules to simply translate a statement in a natural language into some logical construct following simple rules.  One has to understand the semantics of the natural language, and often, context is everything. 
A simple example is the question would you like coffee or tea?  which looks to a mathematician like a simple yes/no-question.  Try answering yes to your colleague next time you are asked this question...
That being said, let's see what your sentence translates to:

Neither L nor S lives in either A or B

The first part Neither L nor S means that whatever follows does not apply to L and does not apply to S:

NOT (S OR L) = (NOT S) AND (NOT L)

The second part in either A or B means that whatever went before applies equally to A and B.

(P is TRUE for A) AND (P is TRUE for B)

So the proposition that X lives in Y, applied to this sentence, means:

((NOT S) lives in A) AND ((NOT L) lives in A)
  AND
  ((NOT S) lives in B) AND ((NOT L) lives in B)

So your friend is right: S does not live in A, S does not live in B, L does not live in A and L does not live in B.

What I think is "One(L or S) cannot live in the places that has been mentioned(i,e A or B) which makes the other one to live in A or B"

Nothing in the given sentence justifies such a conclusion.  If I tell you that Alice does not live in New York, how would that possibly imply that Bob does live in New York?  Nobody ever mentioned that either L or S should live in A or B. 
It looks like you fell for a false dichotomy.  The fact that one thing is not true does not make an unrelated thing true. 

A:

In Standard English, even if a sentence is completely negative in sense, we apply negation only once, at the first possible place.  So, for example, the following sentences all mean the same thing:

None of them ever found any of it anywhere. 
Never did any of them find any of it anywhere. 
None of it was ever found anywhere by any of them. 
Nowhere did any of them ever find any of it. 

All four mean NOT(one of them found some of it somewhere at some point). 
(Note that some of the above versions are awkward; I list them only to demonstrate the grammar and meaning.) 
Your example is similar: it means NOT(one of {L,S} lives in one of {A,B}).  This idea can be phrased in various ways:

Neither L nor S lives in either A or B. (your version)
In neither A nor B does either L or S live. 
Neither A nor B is where either L or S lives. 

In all cases, to correctly interpret the any or either–or or ever or whatnot, you have notice the negation earlier in the sentence. 

A:

Yes, "Neither Nor" and "Either or" can appear in the same sentence. 
Your friend is correct. 
The Neither L nor S prefix means that the rest of the sentence does not apply to (i.e. is not true for) L and also does not apply to S.
Therefore lives in either A or B is not true for L and is also not true for S.
lives in either A or B means lives in A or lives in B.
Therefore, L doesn't live in A, L doesn't live in B, S doesn't live A and S doesn't live in B.

