Those are thoughtful sensible questions from somebody that is taking the trouble to think about what they are and aren't seeing - so it will be my pleasure to give my interpretation. But be aware that my interpretation might well change once I saw all the details of what you are doing, all the scope settings, and fiddled around with the scope.
In image 1, it would appear that it takes approximately 4 microseconds to "Turn the LED on", and 4.6 microseconds to "Turn the LED off". But if you look carefully at the image, there appears to be some ghosting which I am guessing occurs because sometimes the Arduino is interrupting my code with its own code which causes the LED to stay on or off longer than normal. Question: Is the brightness of the ghosting mostly proportional to how often it is occurring or might it only be occurring very rarely but something like phosphor persistence lets me see it?
Quite probably, but there might be alternative explanations.
If it was an analogue scope, the brightness of the ghosts would be proportional to how often it is occurring. But with a digital scope it is also dependent on how the scope "processes waveforms for display". Often scopes will attempt to mimic the effects of "conventional" phosphors, and will succeed to a greater or lesser extent.
An alternative explanation is that there is some "mis-triggering" occurring, possible due to a noise or a poorly choosen trigger threshold voltage. It should be easy to avoid that with this waveform.
Similarly sometimes "auto trigger" can give such a display, but I doubt that is happening in this case.
You might also take note of the large overshoot on transitions, which can be clearly seen in image 2. This is an artefact of your probing technique, in particular using a ~20pF scope probe with a ~15cm ground lead (15cm => ~150nH, resonating at around 90MHz). The only way of avoiding that is to use a much shorter ground pin/bayonet/wire, or by using a lower capacitance probe. I particularly favour "low impedance" or "Z0" probes for this type of signal; commercial ones are excessively expensive, but for these frequencies they are easy to build yourself.
In image 3, I have pressed the “single” button which I assume captures and shows me just one plot. The screen is 800 pixels wide by 480 pixels high and it is still at 60 points, but the image looks way too smooth for just 60 plotted points, what is going on?
This is probably due to the scope "preparing" the sampled data for display by interpolating between fixed points - probably with a sinx/x filter.
Personally I usually prefer to turn such interpolation off, and rely on my eye to connect the individual dots representing individual samples. Other people will disagree - your choice!