I had problems running from an MB102 power module. Using relatively heavy gauge wire and short runs would probably help. Ultimately I built a small board with the LM1117 regulator and a tantalum bypass capacitor very close to the Omega power pins to get reliable operation.
An easy way to test your USB to Serial converter is to wire the Tx and Rx pins together and verify that it can talk to itself. If this is the case then your device drivers are working. When connecting to an external serial device (the Omega) you may have to go Tx to Tx or you may have to go Tx to Rx between the device and USB serial. Labeling of USB to Serial devices is inconsistent.
The Omega should be warm to the touch, but not so hot you can't keep a finger on it indefinitely. Current draw is a few hundred milliamps and fluctuates largely depending upon WiFi activity.
I posted some power data in this thread link text