Jellyfish?

.....
that resemble a 1/4 inch disc, the size of a 1/2
dollar.

from the sound of something that small, I'd think come sort of ctenophore or
comb jelly. Do they have tentacles? If the answer is no, just a couple, or
more like cilia - do a search on the above names. If they have multiple
tentacles, it could be some juvenile jellies or hydroid medusa or something
else entirely

My question, are these animals supposed to show some self propulsion? Or
do they just "go with the flow?"

Yes they do have their own propulsion mechanisms. Many ctenophores do appear
to just "float there" though. Look very closely at them, if there are cilia -
you should be able to see them move. Try looking at them under a black light
or 20,000k, this helps.

Also, will they survive the tropical temperatures of my reef tank? Or do
they need the somewhat cooler waters of the NE?

Depends on species, many will acclimate to warmer waters but takes a little
time. I'd suggest keeping them in water of a similar temp to what they came
from though.

For food, try Artemia nauplii or rotifers.
More than likely, they get eaten by powerheads unless you guard against this.

James