With many sewing machines, it can be impossible to get tension perfectly balanced when using different threads on top and bottom. To get rid of the little dots, you will probably have to use the same thread in both places.
There are different types of invisible thread. I use YLI nylon monofilament and have been able to use it successfully in the bobbin as well as on top. It tends to work better if your machine uses metal bobbins, as plastic bobbins are more likely to distort under the pressure of the thread. When winding a bobbin with nylon monofilament, it's important to wind slowly and to underfill the bobbin, as others have mentioned. Nylon monofilament stretches, and the stretching while winding at high speeds creates problems.
Again, because of the stretchiness of nylon monofilament, it's important to reduce the upper tension when using it. I usually move down from 5 to 3.
For stippling, I would be careful only to do moderate to large-scale stippling. You definitely do not want to do small-scale stippling using nylon monofilament, as it will stiffen up the quilt considerably.