Does your setup show you a shift when you proof with Monitor RGB option?
Yes, it does, and it should, since it turns display color management off. Saturation increase from an sRGB file, minimal change from Adobe RGB, and desaturation from a ProPhoto file. This is all as expected. All this is on a wide gamut display.
As for the proof options: Preserve numbers
checked is equivalent to "assign profile", while Preserve numbers
unchecked is equivalent to "convert to profile". Those two options are drastically different:
Assign profile will show a color shift, since the numbers in the file are interpreted differently. This is an "expert" option that you normally don't use, except when a profile is missing, or to correct outright mistakes.
Convert to profile keeps the appearance unchanged (as far as possible), and changes the RGB numbers to achieve that. This is the normal color management operation that takes place when an image is sent to the display: the embedded document profile is converted to the monitor profile, by the application on the fly, thus preserving the correct appearance.
What really happens when you proof is that color management goes an extra round: Instead of the normal document profile > monitor profile, it goes document profile > proof profile > monitor profile. The first transformation can be either an assign or a convert - that's the checked/unchecked option. The extra round in either case limits the gamut to the proof profile, and that's what you see on screen. But the second transformation is always a convert.
To sum up, think of it this way: Color management always requires two profiles, a source and a destination. If only one is present in the equation, nothing happens. And if the two profiles are identical (by whatever route), again nothing happens and color management is effectively turned off.
Anyway, that's the theory. In your case, if you have Firefox set to mode 1 and you
still see oversaturation, something's not working as it should. That setting should take care of any missing profile by assigning sRGB to it (this is of course all under the assumption that the file has sRGB numbers). I have a wide gamut monitor and Firefox at mode 1, and I never see oversaturation anywhere, with or without embedded profiles. Flash possibly excluded.
In fact that setting displays everything strictly correctly, fully color managed even if there is no embedded profile (since sRGB is assigned). The "normal" situation is that the monitor's native color space is assumed to be "close enough" to sRGB. And with a standard gamut display it usually is, but it's not a perfect match. There will always be slight color and tonal shifts.
Perhaps it's time to run through normal troubleshooting procedures, reset preferences and so on. There is a way to reset Save For Web preferences separately from Photoshop, but I can't recall what it was. For Photoshop it's the standard "three finger salute" on startup. For Firefox maybe a full reinstall and cleanup of user account folders. And recalibrate while you're at it.
Oh, BTW: The preview options in Save For Web are just that - proof options. They don't affect the exported file.
