Many-body states that do not belong to Fock space?
Answers
Probably not a detailed answer, but let's see.
Firstly, you are correct that you can use Fock space to describe interactions, and this is actually mentioned you the (q1) you quoted above. So yes, IF you find a proper basis representing single particle excitations in certain field theories, then you could describe any interaction among them.
The point is, however, what do we mean by a "particle"? The most precise way to define a "particle" in QFT is by finding the poles in the spectrum by somehow calculating Green function "precisely". For example, if you have some non-perturbative approaches for certain QFT, then you can really calculate Green functions and etc. such that, at the end of the day, you could define a "particle" precisely in this field theory, and meanwhile particle number is also well-defined.
But there are QFT that single particles are not well-defined, neither does particle number operator. The most well-known ones of-course are some CFTs, in which no particles are well-defined. Then in these cases, Fock space, which is expanded by particle-number basis, is not well-defined neither.