Both your 5v/1A and 5V/2A are safe and sufficient to run the Omega.
My guess is your powerbank has two separate DC-DC converters for each output. Usually these converters are most efficient when they are used between say 40-90% of their maximum rating . In your case the 5V/1A might be slightly more efficient delivering 0,2A than the 5V/2A output.
For what it is worth: you can always connect a power supply with a higher current rating, be very careful to connect a power supply with a lower current rating unless your hobby is blowing up power supplies.
Output voltage should always be the same as the required input voltage. That said; average power supplies may have a deviation from their rated voltage, like depending on load. Therefore inputs should be somewhat forgiving on deviations from the rated voltage.
The Omega docks with power supply inputs have a DC-DC converter dat make 3.3V out of 5V (probably a so called Low Drop-out regulator, i'll check my dock when i'm home) Probably it kan handle an input range from well over 5V to about 3.5V. The surplus above 3.3V times the drawed current is dissipated as heat by the LDO regulator.
That FAQ you mention also states they ran an Omega off a 10400mAh LiPo battery for four days. Actually this implies an average current drawn by the Omega of around 100mA. I suppose this may vary with the application. With your power bank you may supply the Omega for more than 3 days.
Succes with your projects,
Loek (pronounced as Luke in Dutch)
I tried to identify the DC converters on the dock and the mini-dock, but without succes. The markings do not reveal anything useful in Google.
On the minidock the LDO is the chip with three pins on one side and a big pin on the other.
On the dock is the little one with three pins on one side and two pins on the other (this one had me confused because it has a coil on the side with could indicate it to be a switching DC converter). You may power the dock through the 5V and GND pins on the header as well.
Anyway, as it may come in handy sometimes to power the docks with something else than 5V I kindly request the hardware guys from Onion to reveal these parts, or at least tell us the limits of thes regulators. And yes, beware : when you power the dock with something else than 5V then the 5V pin on the header and USB connector also carries this "something else".