I'll be taking a TGV-Lyria train from Paris to Geneva, arriving at the Cornavin rail station.
Then I plan to connect to a Swiss train from Geneva Cornavin to Montreux.
My questions are:
(1) Is the Swiss train departing from one of the 8 platforms of the Geneva Cornavin station? Or is there a different area for the Swiss domestic trains?
(2) Will there be passport or security check after arriving from France and departing to Switzerland?

Thanks in advance.

The eight tracks are all parallel next to each other. Trains to/from France will depart/arrive from track 7/8, so Swiss domestic trains will depart from any of the other 6 tracks.

Since Switzerland is also part of the Schengen area, there are no permanent passport or controls upon arrival.
Nevertheless, spot controls can happen at any time or due to special circumstances, though not as a daily routine.
While there are also no permanent Customs controls when traveling to/from France (or elsewhere in the EU), Switzerland is not part of the EU so you have the usual restrictions on high-tax goods like spirits, tobacco, etc. like you know them from any international travels. Probably not relevant for you, though, unless you plan to get a new watch or a handbag for 2K in Geneva and travel back home via France

There are no security checks or ticket gates when changing to any of the "Swiss platforms".

The (Milan bound) train Geneva dp 13.39 leaves from track 4.
If you miss it, the next Montreux bound train leaves at 14.12 from track 8

Thank you very much Cowboy and neckervd for your response.