Chain getting in between sprocket and rear dropout My girlfriend has a vintage Nishiki womens road bike, and recently there have been instances when shifting, the chain sometimes falls off the smallest sprocket and ends up stuck between the rear dropout and the sprocket. It has to be forcefully pulled out and placed back on the cogs. Do I need to readjust the derailleur? Are there other possibilities that might cause this to happen?
Yes, you need to adjust the limit screws of the derailleur so it can't pull the chain farther over than the last cog.  There are various videos on YouTube, such as GCN's: