Irreversibility is owing to the large negative Gibbs free energy of the system considered alone, or equivalantly to the large positive entropy change of the (system + surroundings) considered alone. The lower heating value = the higher heating value minus the enthalpy of condensation of water produced in combustion. Enthalpy does not determine irreversibilty --- free energy and entropy do. Endothermic processes such as evaporation of water into dry air are spontaneous and irreversible with negative Gibbs free energy change and positive entropy change even though they have a positive enthalpy change --- the increase in configurational entropy as water evaporates into dry air more than offsets the decrease in thermal entropy.
Yes, I think so, provided that the reactants and products are at the same temperature (environmental temperature), in this case the change in Gibbs free energy is considered as the change in exergy function, which is equal to the reversible work, while the change in the enthalpy is equal to the heating value, the difference between them is the temperature times the change in the entropy which is equal to the irreversibility.
As remarked by Jack properly, the irreversibility means positive internal entropy production, which corresponds to negative Gibbs or Helmholtz free energy productions depending upon whether one deals with isobaric or isochoric systems, respectively. The enthalpy of reaction may be positive or negative which doesn't have any thing to do with whether the process is natural (irreversible) or not.