A few days ago I first noticed these on my Chinese Elm. I think they're tiny bugs but they're so small it's hard to tell. They come off pretty easily with my fingernail.
I tried looking around for possible culprits but nothing seemed to quite match. Do you think they're a sort of scale, or aphid?
Also, I'm moving into a dorm in about a week and will likely take my tree with me to keep inside over winter. Do you think I should instead let it winter at home to guarantee it kills the pests? (but increase the risk of my mom neglecting it)
(Sorry the picture isn't very good. My camera just would not focus on them)

Those are scale insects. Use systemic insecticide, two or three times in a 3-5 week period.

I am not sure this plant would do well over winter indoors. (I have one myself, since about 4 months. I have not decided where it will stay in winter as their frost-hardiness is a heavily debated subject)