What I don't understand is why the signal is dropping to -2.4 or so when triggered then staying below 0.
74HC14 edge is typ. 7ns fast (140MHz), you need a quite fast scope to see real signal.
my scope is a 200Mhz scope, I don't have a function generator that can push it that fast to see if it can actually do it though.
Bandwidth is just a metter of signal attenuation in analog part of a DSO. Sampling rate is a bit more important here.
Let assume your DSO has 1GSpS (1 sample per 1ns). You will get less then 10 samples to reconstruct a signal from. Depending on interpolation function your scope is using, you may get ringing on the screen which is actually not in real signal.
Dave has a vid on this topic.