Hi,
There is no way to charge the phone. Wireless Charging, Powered USB, OTG cables..... tried it all, it doesn't work. See:
http://support.displaylink.com/knowl...isplaylink-for
About the screen issues, I am using "SecondScreen" and "Pie Control" apps to overcome some of the issues. The only problem is: your phone has to be rooted to use SecondScreen.
Android + Displaylink works, but it is not ideal (yet). Maybe a Microsoft Lumia 950 is a better solution for you. Or just wait till "docking your smartphone" becomes mainstream. It's still in a testing phase.