I have never used the Gold or Silver card as egates, always issued a separate one. Nowadays you can't even do that and residents/citizens have to enable their National ID as their egate.
To answer the questions:
Yes all 34 visa waiver country citizens can get egate cards issued at Dubai immigration counters only, Abu Dhabi will not issue them. There are counters located in Terminal 1 (upstairs to the food court, before check-in), at T3 F/J check-in area (after immigration
, if the window is not manned, walk round through the door on the left and
there will be an officer slouching at his desk looking for enthused) and before immigration at arrivals. I think there is one more at T3 Y area but can't remember where. In addition, there are a few other counters in the city.
The card is valid for 2 years from date of issue and costs AED 210 (last I checked).