1. Here http://www-geant4.kek.jp/lxr/source/processes/hadronic/models/neutron_hp/src/G4NeutronHPInelasticCompFS.cc#L327 maximum energy for scattered neutron is defined as E'=E+QI (where QI < 0) but ENDF-6 Formats Manual states for MT=91 that 0 <= E' <= E-U (where U is the threshold energy). See chapters 5.2 and 5.3 for details.

2. Here http://www-geant4.kek.jp/lxr/source/processes/hadronic/models/neutron_hp/src/G4NeutronHPInelasticCompFS.cc#L340 level increment could lead to further deexcitation via photon with energy greater than incident neutron energy. Is it a feature?

3. ENDF standard say that photons from the continuum should be present in MF=12 - MF=15. In case there is no such data it may be an idea worth considering to add at http://www-geant4.kek.jp/lxr/source/processes/hadronic/models/neutron_hp/src/G4NeutronHPInelasticCompFS.cc#L416 something like:
if ( !(G4NeutronHPManager::GetInstance()->GetDoNotAdjustFinalState()) ) thePhotons->push_back(theRestEnergy);
This will implement the treatment of continuum in the same manner as adopted for discrete levels.

Or even better add a specific function in G4NeutronHPManager which allows user to skip gammas from deexcitation of continuum - all of them or only gammas which are not from discrete levels.