To help with your questions, I can honestly say, I hadn't even thought to look for such an option. One of my older phones had it, but really, I don't see a need for it. The OS does a good enough job killing apps on an as-needed basis. Unless you are just talking about like when you have a ton of notifications up there, like for completed downloads, new mail notifications, etc. Pretty sure if you click the little X on the pulldown for the notification bar, that it will delete all of the notifications. It doesn't kill the apps that are running though.
Finally, as far as apps/widgets to recommend... that is a tough one.
I use a lot at different times. Actually, on my One, I haven't really put a ton of widgets on, just apps. I of course have to have my PowerAmp widget for playing my music. I use the GSAM battery widget, and also use the toggle widgets from Beautiful Widgets for things like Brightness, turning the Wifi, bluetooth and Mobile data on and off, and also to replace the clock/weather. Haven't had a need for a ton of widgets yet.