What is the relation between First Order Logic and First Order Theory?

Question

I thought that any FOT is a subset of FOL, but that does not seem to be the case, because FOL is complete (every formula is either valid or invalid), while some FOT (like linear integer arithmetic) is not complete.

So, is FOL more expressive than any of FOT? Or incomparable?

Also, the statement "there are statements that are valid in LIA but cannot be proved using axioms of LIA" is weird. How can the statement be valid if we cannot prove its validity? I always thought that if you cannot prove the validity of the statement, then you cannot claim it is valid.