@kugiigi I see, I haven't used UT in a few months, so I might be missing a few things.
In that case, my suggestions are to add minimize functionality to the apps in the side bar, because that way it will replace the "background apps" screen. That in turn will solve the issue about going quickly to the home screen.
And if apps are allowed to be added to the home screen, that will make the home screen feel like one. It will also separate the home screen, side bar and the app drawer clearly.
My suggestion for the lock screen is to make it look like Android. It's simple and it works - a clock, date and notifications.
In addition, the background apps screen could be improved too, along with my other suggestions. If its opened with a swipe up it would be more intuitive IMO. The layout could resemble the tab layout of Firefox on Android - simple grid with an "x" above each app to close it. And finally, a button to close all apps.