What I'm really trying to determine here is if this is normal for a new scope to be unable to trigger properly off its own test signal after pressing "auto". I know the "auto" setting on a scope if far from perfect but if it was ever supposed to work correctly I'd assume it would be with its own test signal.
No, it is not normal. It does not necessarily mean the scope is broken, but something is wrong, either in the test signal, the probing, the scope, or perhaps the nearby RF environment.
Compare your pictures to Stupid Beard's. Notice especially that your traces show much "fatter" width on the flat portions of the waveforms. That indicates that you've got some high frequency noise superimposed on the 1 kHz test signal. You can see the noise better when you used the faster timebase. Just eyeballing the fatness of your traces, it looks like around 1/4 volt or so.
The task becomes figuring out where the noise is coming from.
You might try setting the trigger to "single" mode, and running once, at a very fast timebase, fast enough to clearly determine the frequency of the noise. Does the noise even have a regular and consistent frequency? Are you very near a powerful radio transmitter? Is the frequency of the noise close to the frequency of the radio transmitter?
It also might be interesting to try probing a known smooth source, like a battery. Do you see that same sort of fat trace with noise?
Another experiment might be to take the scope to a different location. Maybe try a friend's house, or at least a spot as far away from your normal workbench as practical in your home. If the noise disappears when you move the scope away, the noise could be coming from other equipment in your lab.
The noise could be coming from something inside the scope, in which case you might have an issue to be dealt with by sending it in for warranty repair.