Changing name does not propogate to iPhone
Changing the name on the Web-version of Duolingo does not propogate to the iPhone App. In fact. The app itself seems to be acting very strange and crashing after changing name.
Now I changed my name back again on the web and the App started working again.
If you change your username and go to the app you'll be logged out (as you experienced) and need to re-enter your new username along with the password you used before. Once in, you should be good to go.