Why am I getting Universal Clipboard popups when I'm not typing anything?
You are likely correct. If you get a UI that says that the system is "Pasting from [my MacBook's Name]", then this likely means that the application in the foreground of your iPhone is reading from the universal clipboard.  It is unlikely that it is being read by another program in the background. So it is likely that Weather Underground is attempting to read from the Clipboard.

See:
* https://support.apple.com/guide/mac-help/copy-and-paste-between-devices-mchl70368996/mac

My suggestion is that you test by doing the following:


Repeatedly copy something on your macbook, kill the Weather app, then exercise the weather app. 
Take your Macbook off the network and repeatedly run and kill the Weather app.