Do the dots look like salt grains(that would be Ich, like DQ suggested)? Are they raised? Fluffy-looking in any way? About how large are they? Any unusual behavior?
I'm no expert in diseases by any means, so someone else will have to pop in and tell ya what to do.....but without a picture, as much info as possible will help greatly.
To offer some other advice....you really should get your guy a heater. Bettas are tropical fish who need and prefer temps of 78-82 degrees F. Lower temps make them lethargic(and prone to disease)and fluctuations in temperature(which it seems is whats happening if you have a lamp as your only source of heat)will stress them, making them prone to disease as well. You'll want to get a heater ASAP and get the temp stable, that will help loads, especially if we're dealing with Ich.
Also, a 2 gallon should be cleaned 2-3 times a week, even more often if you think your guy is sick. Sometimes, all a betta needs is super clean, warm water.
As for light, bettas don't really need a light/lamp, most just used them for viewing purposes or a temporary heating source until a heater can be obtained.