I have a solution. It's ugly, and it's not overly practical, but it works - so if you're reading this and returning / replacing is not an option... here's your out.
I got mine about 8 days ago and have noticed the same things described here. At first I thought it was something I'd done to it when I rooted it, but after seeing all this, I'm looking into returning it and/or getting the screen fixed.
I did discover a few interesting things through trial and error. I was originally using AOKP's ICS rom when the problem started - thought there was something wrong with the rom, so I tried the Caulken, CM9 Alpha 6, and originally ODIN'd back to the factory settings, confirmed issue is happening across all of them. Happened less on factory, but I suspect that's due to the modem it was using giving me better reception out here in nowheresville where I live.
I also confirmed that switching it over to Airplane mode stops it - but then you're in Airplane mode... so yeah.
Next, after re-rooting and re-ROMing, I downloaded an antique app (not on android market, this one I found in the wilds of the internet) called ButtonRemapper - icon is a green circle with a puzzle piece in it. From it, I remapped the hardware button for Menu to something completely useless - the "Focus" command. Since it's only active when you're taking a picture, it basically turned the hardware button off in normal use mode.
The good news is... by deactivating the Menu button, I stopped getting "phantom menu" presses - which also confirms 100% this is a HARDWARE problem. Anyone who wants to take this fight to Sprint/Samsung - there's some ammo for you.
The bad news is... I started getting phantom Search, Home, and Back presses once I'd blocked Menu. I wound up going back to my trusty ButtonRemapper and also setting all those buttons to Focus, which disabled them all as well.
Now, to complete things, I went into AOKP's settings (though other robs with this option should work as well) and activated the "Menu" button bar, giving myself a soft menu at the bottom of the screen that replicated the commands of the now-defunct hardware buttons - just like the Galaxy Nexus.
Top off with a strip of white or black electical tape to cover up the damn LEDs for your dead buttons. Lather, rinse, repeat.
ButtonRemapper ->
[App] ButtonRemapper - Beta v0.2.4 (New Galaxy S version added) - xda-developers