Thermodynamics - Q, U, W and H

I have understood that Q=heat absorbed, ΔU=energy change, ΔH=enthalpy change, W=work done by system = Δ(PV)+W_mech (where W_mech is mechanical work, P is pressure and V is volume).

What I don't understand is what the actual, exact relationships are between these values (which hold constant regardless of the conditions, whether we are changing pressure and volume at the same time, or whether or not there is W_mech; these complications aren't shown in any textbooks I have). First I'd like to look at these actual relationships (i.e. how to calculate each value from the others, without any approximations being made about the conditions, e.g. "pressure is constant") and then I'd like to consider what we can say under these specific conditions (e.g. "volume is constant" or "pressure is constant").

For example, I've heard that ΔH=ΔU+PΔV. I take it this is ΔH=ΔU+P(V2-V1), which would suggest PΔV is the work done by the system in increasing volume. So this make W=Δ(PV)+W_mech the work done by the system, but how does that relate to the other values? In my expression for ΔH, what if pressure is changing? What if there is mechanical work? And what are the other similar relationships? Where does Q come into it? All of these questions!