emacs-pgtk for WSL2 - identification and clipboard issues

Question

I installed emacs-pgtk version 29.0.50 (build 2, x86_64-pc-linux-gnu, GTK+ Version 3.24.33, cairo version 1.16.0) from the git repo following these instructions with

./configure --with-native-compilation --with-xml2 --with-json --with-pgtk

Shortcut on Windows Start menu did not appear so I manually created using this suggestion (essentially creating a shortcut to the location, C:\Windows\System32\wslg.exe ~ -d Ubuntu-22.04 emacs).

Overall it looks to be working okay but some strange things.

1. Clipboard: I am unable to paste content into Windows applications after using kill-ring-save and kill-region in this emacs-pgtk that I launch with the shortcut. However, kill-line works as expected and I can paste content into Windows; and kill-ring-save and kill-region work if I launch emacs-pgtk from the WSL2 Ubuntu terminal. Why would this behavior be different?

2. Is there a way to identify that this is a Linux GUI app in emacs? I already have font specifications for emacs-pgtk opened from the WSL2 Ubuntu terminal, but the I need to specify a different font size for the Linux GUI app. The system-type (gnu/linux) is same as if I launch from the terminal so I cannot use that to distinguish the GUI app.