So, I think different books define it differently. Depending on how you define work in the equation affects what sign you put on it. One thing that is universal is that internal energy increases if heat is gained by the system or if work is done ON the system and that internal energy decreases if heat is lost or if work is done BY the system.
For this reason, I think it's better to define changes in internal energy like this:
Change in internal energy = (heat gained- heat lost) + (Work done on system- work done by system)
Or, you could also define it in its most basic terms like change in internal energy = energy gained - energy lost. In this case change in internal energy = (heat gained + work done on system) - (heat lost + work done by system)
Also, I think you're right when you say H = PV + U is analogous to E = q + w because the heat content doesn't really change, only gets converted between work and internal energy, just like energy is conserved and just converted between heat and work in the other equation.
Pretty cool is if you consider that U = q + PV, then if no work is done U = q or internal energy change = heat change. Then substitute it into enthalpy equation(H = PV + U) you get H = PV + (q + PV), where if no work is done then both the PV's drop and enthalpy change = heat change. Similarly, in the other equation E = q + w, if you assume that no work is done than change in Energy = heat change also.
Conceptually, I think it also makes sense that an exothermic reaction would lead to a decrease in internal energy of the molecules, because exothermic reactions are more likely to be spontaneous(depending on temperature and entropy change) and in physics and chemistry molecules or what not always want to exist at the lowest possible internal or potential energy.
Hopefully, that helps and I didn't mess anything up. This stuff was a headache to learn, and I think I only got it about a week ago.
