In dump1090 what's the criteria for displaying tail #s etc?

It looks like you are having similar idea to others. Check out feature requests.
feature-requests/

Here, they’re generally squawking 1200, which makes sense for touch-and-go & etc. But when I select that line in dump1090, only the ICAO24 address is shown (for example AE065C)

Planes don’t send their tail number in the transponder messages. Some planes do set their Ident to their tail number.
There is a way to mathematically compute the ICAO24 to tail number and this is a requested feature that in our to-do list.

If I click on flightware, it takes me to airframes.org, where I can lookup that this is KC-135 tail number 59-1507. Why doesn’t dump1090 manage that? And why doesn’t Flightaware
manage it instead of having me type it in and do the lookup manually at airframes.org?

Yup. Another feature on our to-do list.

Finally, why do I never see positions, (especially mlat positions) on VFR (squawking 1200) flights? In the case of flights in and out of KFOE, I know there are several PiAware users in the >> immediate area who have far better numbers than I do, and I would think mlat would be easy. Is it that the system just doesn’t do it for VFR flights?

By default we only show planes flying with a flight plan. You can go into your account options to show “position-only flights” and see VFR planes.
ads-b-flight-tracking-f21/what-qualifies-as-a-flight-for-the-purposes-of-t38262.html