LaTeX.org

I can describe only what I can see on my Macbook with LyX2.4.3. There is no Date setting in the Preferences settings on my machine.
The handling of date fields had been changed since LyX2.4.
I use the Dialog "Field Settings". It can be reached via the menu by clicking "Insert>Field>Other...".
Not ...

posted in error

LyX stores the generated PDF files all at a fixed position, the temporary directory.
You can view that folder from the "about LyX" dialog and get the path to that folder as well.
You can change the location of that temporary directory in LyX / Preferences / Paths / Temporary directory to whatever ...

I don't have a good suggestion. I have heard about people having problems with folders in a path being synchronized with one-drive or other cloud software.

The files created during the "compilation process" and the resulting PDF file are always stored in the temporary directory. This temporary directory is separate from the doc lyx file and the path can be seen in Lyx Preferences or in the About Lyx Dialog. I run LyX 2.4.2.1 and the about LyX dialog ...

I have the same problem on my macbook since LyX 2.4.2.
The bug is not unknown. See https://www.lyx.org/trac/ticket/13120
I hope for a fast solution.

I had a similar problem with MacOS Sequoia.
MacOS Sequoia appears to take the shift-command-M key binding for itself. This prevents it being used as a shortcut to enter a display maths formula.
The binding can be released by unticking "Open man Page in Terminal" in the MacOS Settings -> Keyboard ...