[big]

As I understand it,  :delta: H = q at constant pressure (and also assuming that no non-expansion work is done).

Then, I came upon an example problem in which an ideal gas is "expanded adiabatically against a constant external pressure of 1.00 bar." Because the process is adiabatic, q=0. However, the :delta: H turned out to be -2.96 kJ. Why isn't :delta: H = q if the gas is expanding at constant external pressure? Does this mean then that :delta: H = q only at constant internal pressure, since internal pressure is changing still?

[Yggdrasil]

Does this mean then that :delta: H = q only at constant internal pressure, since internal pressure is changing still?

  Yes, what matters is whether the internal pressure stays constant, not the external pressure.