I will periodically close any apps that I'm done using via the overview screen. Occasionally, I will see several apps open that I did not start, but had open days ago. Is this a bug or a virus?
If one thing for sure has improved with successive Android versions, it is memory management
It's likely that the apps you see after closing are apps which have a schedule (like scheduled scan) that wakes them up or are apps that get triggered on action like taking a photo and Dropbox gets activated because auto upload of photos is enabled. Look into that aspect.
It doesn't sound like a bug (incidentally, even after reboot you will see apps , unlike pre lollipop versions)
By swiping you aren't killing the app
See What actually happens when you swipe an app out of the recent apps list? which discusses at length what happens when you swipe an app
Actually, removing an entry in recent tasks will kill any background processes that exist for the process. It won't directly causes services to stop, however there is an API for them to find out the task was removed to decide if they want this to mean they should stop. This is so that removing say the recent task of an e-mail app won't cause it to stop checking for e-mail.
Also from Android Developers you can see that an app can generate multiple views in recents and can even retain finished tasks