I ran into the same trouble as #2 a while ago. I was running a nav map and trapster (along with gps of course). It got damned hot and was not keeping itself charged. I have not repeated the experiment.
Wall power and DC adapter power quite often ARE different. Not in voltage, but in wattage. Some DC adapters supply a lot less wattage than a direct wall connection.
That's why I'd love to have a dock that had both a USB port for the computer and a separate power connection. I've had other 3rd party docks that did this for other phones. It'd be great to have one for the D2.