I believe the Addon Manager will prompt you to switch to Windowed mode anyway, if you already aren't.
I can't reproduce what you are describing. Are you sure the dialog disappears as soon as you press the "Register Serial" and not, instead, after you typed it and inserted it ? Because, if the sim goes back to flying without any message, after you insert the Serial Number, it simply means you either entered it entirely wrong or, you entered it correctly, but for the wrong product.
The Reinstall (which, as explained on the install guide, it's really called "Server-assisted Reinstall" but that didn't really fit into the button...), only works if you purchased using the integrated purchase procedure, not if you purchased from the web page.
If you purchased from the web page, the one and only option is to use the "Register Serial" feature.
If the problem is not happening after inserting the Serial number, then it must be other software interfering. If you use the FSX to Prepar3D Migration Tool, be sure you follow the instructions in this thread:
http://www.fsdreamteam.com/forum/index.php?topic=7237.0