I have a Chinese elm bonsai. I made the mistake of keeping it inside the past year, so I moved it out in December so it can experience dormancy. It has been in a shed with a window door in a styrofoam box (lid open) with a heating pad to ensure the roots don't freeze. The heating pad has a probe so the temperature does not get too high. On particularly cold days, I also wrap up the pot. On warmer days, I move it out of the shed. I water it every couple days, which is usually when the soil is getting dry. This method seemed to have worked so far, but the last few days the leaves turned brittle and brown. What should I do? Is there any hope of saving it? (The photo makes the leaves look greener than they are.)

It is winter and the elm is a decidious tree. It will stay dormant for a few weeks more. If it lives or not you will see in a couple of weeks when it breaks dormancy.. Meanwhile, just relax.

You mean the leaves of the Chinese elm bonsai are supposed to turn brown, brittle, and fall off easily during dormancy? Why is it occurring now, about a month and a half later? 

Yes, decidious means that it sheds leafs in a yearly manner. In my experience chinese elms are late to drop, February in my conditions.

What about the spots? It may be a little difficult to see in the photo, but some of the leaves that are not brown and brittle yet have white splotches on them. Is that normal, too? I read that those could mean pests or lack of nutrients. 

If the plant is dormant then it does not need nutrients. The leaves are dying so in quite a short time will all drop. Do not allow to dry out even without leaves, otherwise keep it frost free and do not keep moving it. Look at the leaves, if there is anything on the leaves it will be underneath, pull gently any leaves that are dead or nearly.

It is also in desperate need of pruning, you can do it once all the leaves are gone. 

When I move my bonsai between the outside and the shed, there is no difference in conditions. The shed is basically a wooden box, and I only put it in there so the winter winds do not dry it out or contribute to root temperature drops.