Your little bonsai is a juniper tree.
I can't tell from the picture if it really is mold or not, but if it is, you are right it is a sign the soil is staying too moist. You said you water until the water runs out the drain holes, which is the right thing to do. But your bonsai pot is sitting in an outer container. Do you get rid of the water from the outer container? If not then, the soil is sitting in water and staying wet too long.
I can't see very much of your soil, but these little mass produced juniper cuttings are often sold in regular potting soil, which is very organic and moisture holding. What it needs is bonsai soil which is usually something like small gravel with a bit of fine pine bark, very inorganic, loose and free draining. Now is the wrong time for repotting, but in the spring you should be sure it is in good bonsai soil.
It wasn't bad what they told you about judging by when the pot is light, but I disagree with "it should always be heavy." In general bonsais should dry out a bit between waterings. Here is some bonsai information:
https://www.helpfulgardener.com/bonsai/ one of the articles is about watering.
The other thing you need to know is that your juniper cannot live inside. Junipers die indoors. They are a temperate climate tree that needs winter dormancy. In the warm dry environment inside, they dry out (not from lack of water at the roots, but from the foliage drying out) and they exhaust themselves from growing all the time without dormancy. It might or might not die the first winter, but by the second with no dormancy it is pretty guaranteed to be totally used up.
You didn't say where you are, but yes it LIKES cold winters. It is hardy well down in the negative numbers.
If you get lots of snow, it would need some protection from snow pack, but it needs to stay cold.
