Hey,
My app wants a tray icon that stays active even when the app is closed. (Windows)
All solution i found for this seem to involve Forms which does not work with unity3d.
My questions are this: How do i create it, how do i add notifications to it, how do i add a menu?
Unity has no API for system tray icons, as those are highly operating system specific.
Perhaps there is something on the asset store you can use? Google over there for things like system tray, etc.
The general solution for things like this is for you to write a native plugin that you interoperate with to do what you need.
I would start with getting a basic native plugin going so you understand the interop process, then start working through native code tutorials for whatever operating system you are using.
Its sad unity lacks operating system specific features. For Windows it would be good to have WMI and Forms.
Right now the best solution might be to make a small extra app that only maintains the tray and related updating with a different engine.
Write a plugin, sell it on the asset store, make millions.
If you can’t make millions, Unity probably shouldn’t bother with the feature either. 
I had the feeling unity wants to grow into 2D market an maybe also more into daily use apps instead of only games. Tray icons would help certainly for that, but also there is the general thing of being more featurecomplete: I think os specific classes and extensions really would improve Unity and attract devs.
For writing my own plugin i dont know the windows api and stuff good enough to make a tray icon and it feels like i have to reinvent the wheel.