I just received my vessel sink and now need to purchase the right drain... The actual opening measures 1 3/4" and the recess for the flange/trim is 2 3/4". The various drains I've been looking at measure 1 1/4", 1 1/2" or 1 3/4". Am I limited to 1 3/4" drains or can I use one smaller? Also, my sink has no overflow so I definitely need a drain with no overflow?

On those you need to follow the manufacturer's recommendations. Home depot expo carries a line of drains for those sinks.