Q. Do you have to run the program in the background so it will index new files?
No. Everything updates its database when you run it.

http://www.voidtools.com/faq/#Does__...system_changes
Does "Everything" miss changes made to the file system if it is not running?

No, "Everything" can be closed and restarted without missing changes made to the file system (even across system restarts).

"Everything" updates the database when it is started.