To edit Cortana settings, press the Search button (or launch the app) hit the burger in the upper left.
Select Notebook, from there settings. TAP IT ONCE, it take s a while to load, but then you can set the settings to use above lockscreen. Still doesn't work with the screen off though.
For those with battery issues do the following:
Turn on Battery Saver, by default it will block all chatter. Go into Battery Saver from All Settings > System > Battery Saver
Tap on Battery Saver Settings at the bottom
Tap Add an App, add apps one at a time to find out which one drains your phone, uninstall and re-install said app.
Additionally there is a lot of syncing going on. You might want to turn off OneDrive uploads, and theme syncing for the time being. Adjust your sync schedule in the outlook app for you e-mails. One other thing is to check the feedback/diagnostic sending. It could be sending a bunch of telementry data back to MS for diag (this is why resetting after upgrading is bad, it all gets lost..... unless your phone is like really bricked)
Finally go to store and update all the apps possible. I have battery drain but its a work phone so there is a crap ton of stuff I do on it daily. It is however enough to notice, according to Battery Saver 89% of usage is in my screen.... I think turning off auto brightness might help as well but haven't confirmed it.
Side Note: My MS Surface Arc BT mouse works like a charm, however I learned that the soft navigation keys are necessary, there is no way to go back or home or search without them. The capacitive keys are not accessible via the mouse....