The title says most of it...not sure whether I should set Photoshop to Monitor RGB color profile and set "show proof colors" to true when working on desktop layouts. I notice that Firefox uses my monitor's color profile and Chrome does not. If I select sRGB everything seems way off, especially compared to what I see in either Chrome or Firefox.
If it's any consolation, I am using two identical Dell IPS monitors (2209WAf). I have not calibrated them, yet, other then adjusting brightness/contrast based on color bars and other test charts (I know how to calibrate a video monitor off of SMPTE color bars, not saying I'm a pro at it, but I know these are close).
Just so you can see the dramatic differences between the 3 color profiles - here's a shot comparing the three (taking a screen shot in photoshop, cropping it down, saving as JPEG)
The 2209WA is an sRGB monitor - it is not "wide gamut" (good monitor by the way - I have one) - the color space native to it should be sRGB, or as close to sRGB as it can get (and it does pretty well according to reviews).
So, theoretically you should be able to use sRGB all the way through without needing to do any colour space conversion or having a monitor profile at all.
If you do use a monitor profile, feel free to use it all the time no matter what medium you're designing for - but you'll only really benefit from it if it's properly calibrated to your actual monitor. And it may be worth keeping in mind that if your final target medium is the web, 99.9% of people who view it are probably going to have worse colour accuracy than your 2209WA.
In any rate, if you are designing for screen (and you're not using some wide-gamut graphics setup) there is no reason to use the "Proof colours" feature of Photoshop. Proof colours is for when your final output will be to a different medium, such as for print, and you want to momentarily preview what colours you'll lose as a part of the different colour gamut of the output medium. Even so, this feature is intended to be normally off while you are designing and editing, to get as accurate as possible a view of the colours as they actually are in the image, and only turned on momentarily to double-check what might happen to them when the image is printed (or output to your chosen final output medium). If your output medium is the same as the colour space your image is in (or if you have no control over what equipment prints and/or displays it) - then proof colours isn't for you.
By the way, this is a good image for previewing both gamma and colour space coverage - your own test image is mostly greys and highly saturated blues, neither of which should alter much with different colour profiles. Note that you probably shouldn't compare this sRGB rendering of the Macbeth Colorchecker with a printed version of it.