Are there situations that a crashed Android app's task will remain running in the background?
If the app crashes the app will restart most probably by default and continue in background.
You cannot run multiple instances of the same app in android.
The clear memory button is used to free the memory occupied by all open apps and the apps will restart.
Don't do it frequently because the apps will occupy the space again and it will take more power and processing power to free the memory and reload the apps.
You don't need to be concerned with resource usage, android os is smart enough. Just make sure you don't keep unnecessary apps on your phone and huge media files that are not required.