In principle, you can get all the information you need on the FLUKA
neutron cross sections by setting a printing flag in option LOW-NEUT.
From the manual:

     WHAT(4) = printing flag: from 0.0 to 3.0 increases the amount of
               output about cross-sections, kerma factors, etc.
               Default: 0.0 (minimum output)
 
However, the information is given in a form that is difficult to understand
unless one is familiar with multigroup neutron transport codes. Anyway,
one will get a table of which a few lines are reproduced here:

1                     CROSS SECTIONS FOR MEDIA     1
                      (RESIDUAL NUCLEI INFORMATIONS AVAILABLE)
 GROUP    SIGT    SIGST   PNUP   PNABS  GAMGEN NU*FIS   EDEP
DOWNSCATTER MATRIX 
          barn     barn  (PNEL    PXN   PFISS  PNGAM)  GeV/col
    1 5.826E+00 9.287E+00  .0000 1.5939 1.2904  .3886 1.536E-02   .3228
.0079   .0021   .0023   .0020   .0013   .0011   .0035
.......................................................................

   26 6.861E+00 6.697E+00  .0000  .9761 1.0609  .0181 1.458E-03
.......................................................................

Explanation of the relevant quantities:

Group 1 (the highest): Total cross section (SIGT) = 5.826E+00 barn
                      "Scattering" cross. s. (SIGST) = 9.287E+00 barn
                      Probability of Non Absorption (PNABS) = 1.5939

Group 26             : Total cross section (SIGT) = 6.861E+00 barn
                      "Scattering" cross. s. (SIGST) = 6.697E+00 barn
                      Probability of Non Absorption (PNABS) = .9761

The data for the first groups will probably look strange (scattering
cross section larger than total cross section): the reason is that there
is neutron production through (n,xn) reactions (fission is accounted for
separately), and here "scattering" means "number of outgoing neutrons
times cross section" or also "changing energy group". Since
more than one neutron on average is exiting a collision, the probability
of non absorption is larger than 1.

But looking at the lower groups (group 26 has been copied here as an
example) one will see that the data make more sense. The absorption cross
sections (all of them included) will be = total - scattering = 6.861 -
6.697 = 0.164 barn

The neutron balance according to your definition is related to B. Is the
term that enters in the diffusion equation for a steady in time solution.
The diffusion equation is described as:

d(Flux(r))/dt = D*\Delta2(Flux(r)) + S(r) - \Sigma_abs*Flux(r)

where S(r) term describing the sources and \Sigma_abs the absorption
cross section.

With words the diffusion equation is (per unit volume):


dFlux(r)/dt = -[outflow rate] + [production rate] - [absorption rate]


and for a steady state solution


[outflow rate] = [production rate] - [absorption rate] = neutron balance


The particle 8=neutron is scoring the neutron fluence, the quantity most
frequently used for describing neutron fields. Fluence is defined as the
number of particles that penetrate a sphere with a cross section of pi*r^2
= 1cm2 per unit of time and/or energy. Otherwise, the particles crossing a
surface of 1cm2 that is ALWAYS perpendicular to the direction of the
particle.

In order to kill all low-energy neutrons below the group transport
boundary one should use LOW-BIAS with WHAT(1) set to 1, i.e., selecting
the highest energy group as cut-off boundary (inclusive). If you want to
select a even higher neutron cut-off PART-THR has to be used
which will then also stop the low-energy neutron group transport.

Example:

no neutron transport below 19.6 MeV
LOW-BIAS       1.0       0.0               Reg1     Reg2

no neutron transport below 500.0 MeV
PART-THR      -0.5   NEUTRON

Yes, there are low-energy neutron cross section data sets available in FLUKA
which take into account molecular bindings. The cross section data set has
to be associated with the material by the card LOW-MAT. See Chapter 10 of
the manual for a list of available data sets.