If I go to Rovaniemi in October to visit the Santa Claus Village, is this a time when it is common to see the Aurora Borealis and maybe some snow?
The best time to visit is around Easter, when there's lots of snow, it's not too cold and there's sunlight, but October is a reasonable option.
Snowfall: Statistically, the "first snow" (ensilumi), meaning uninterrupted snow cover, falls around the 2nd week of October. However, this is highly variable and there's a real chance of slush or rain, especially towards the beginning of the month.
Aurora: Statistically, October is the third best month for spotting auroras in northern Finland, behind only Jan/Feb. (Rovaniemi is about halfway between Ivalo=I and Oulu=O in this graph.)