flightaware.com/live/airport/KDMW
especially N227KM which seems to have arrived and departed from the same airport at the same time.
That’s correct. The flight starts and ends at the same airport, so you’ll see the same information on the arrivals and departures boards.
Given the number of flights that aircraft is making on the same “trip” (starting at A headed to B, diverting back to A), I’d guess they’re training flights.
We call this a “Touch-n-Go”. Upon landing safely, during the roll-out we raise the flaps, apply full-throttle and takeoff again.
Okay, I’ll give it a shot.
As far as the flights themselves are concerned, there is definitely something that seems out of the ordinary. For whatever reason, the aircraft made 3 attempts to fly to KCKB, and diverted back to KDMW each time. As far as arriving and departing at the same time, I think you’re reading the charts wrong. Yes, the same flight appears on the “Departures” list and the “Arrivals” list. This is because the plane took off and landed at the same airport (KDMW). The departure and arrival times are consistent with one another, so there’s no problem there. The only problem I see with the information itself is in the last (7:55p.m.) flight. If you look at the log, you’ll notice that there’s a “jump” from 8:22p.m. to 12:08a.m. The info. FlightAware receives comes directly from the FAA, so I would imagine there’s some sort of glitch on their end.
The real questions are 1) Why does this plane continue to divert back to KDMW? and 2) Why does the pilot seem to love filing flight plan after flight plan, when a simple “diversion” would suffice (and make it much easier to track the plane as well).
I hope this helps a little. Daniel or Mark could probably shed more light into it, and they use the proper terms when they explain things. 
I guess I need to type a little faster. Beaten twice! DAMN.
BTW: No touch & go, Needle. The t/o and landing times aren’t the same. 
Mark Duell:
Any idea why the times are so screwy on the 7:55 flight?
Could be a long IFR flight where they cancelled IFR and went VFR for a while, or two different IFR flights. Either way, we never saw a cancellation/arrival for the first segment or a departure for the second segment.