Had all these issues. This is what I had to do to sort them out:
No Notifications: Uninstall the app and then go to Twitter.com. In Applications, remove EVERYTHING. I tried only removing Windows Phone but for some reason that did not work. So since I couldn't pinpoint what I had to remove, I removed everything. Then reinstall the app and log in. That should do the trick.
Multiple notifications: Ben "The PC Guy" said on Twitter you just have to uninstall/reinstall. That's what I did and it's been working fine since.