Zoom desktop client software stores any temporary files?

If you don't have local recording turned on, then video files won't be saved locally, but a chat transcript will be saved for the host.
It will be saved to the "documents" local recording storage location:

• Windows: C:\Users\[Username]\Documents\Zoom
• Mac: /Users/[Username]/Documents/Zoom
• Linux: home/[Username]/Documents/Zoom

https://support.zoom.us/hc/en-us/articles/115004792763-Saving-in-meeting-chat

There is a local location that the Zoom desktop client uses, but I don't see any temporary directories there that would affect cloud recording. If you wanted to test it to verify, you can target these Zoom background file save locations:

• For Mac: HD/Users/username/Library/Application Support/zoomus
• For Windows: C:\Users\USER_NAME\AppData\Roaming\Zoom

Where are temporary files stored when zoom is updating? Windows machines

.exe file for client installer

That should be the location where the user has their downloads set to.

When you file transfer through the chat, does it write any temporary files during the transfer process? Not talking about actually downloading the file to my local PC. Wondering if when the document gets uploaded to the chat and becomes visable to me, are there temporary files written on my local PC prior to me actually downloading the file?

You could try (on Windows) downloading the SysInternals Suite from the MS Store, using the file monitor to see what the client does when and after it is recording.  Am pretty sure it doesn't save anything, but to be sure you should try watching its behaviour!