Don Mueang might need money spending to make it capable of handling the A380 but B777 would be possible. Suvarnabhumi is operating above design capacity at the moment and Air Asia got kicked over to DMK.
Operating from two different airports in the same city increases costs for an airline, EK operate from Heathrow and Gatwick because of severe limits on slot availability at LHR though there are enough LGW flights to make it worthwhile having the staff there and operating premium passenger lounges. The catchment area for passengers may be increased and flight timings conveniently spaced out between both airports to provide a wide choice of departure times.
I wouldn't be worth the additional costs to operate from another airport just for a single daily flight and I'm sure Emirates have enough money and "influence" to get the slots they need at Suvarnabhumi which has more convenient connections and transport links to the city.