You should always embed the profile. With the notable exception if Internet Explorer (which seems to be a lost cause), most browsers today handle color management well. At least Safari and Firefox both do.
Photoshop's Save For Web still strips the profile at default settings, so you need to check the "Embed Color Profile" box. That setting sticks. Personally I think it's well past time to change this default, but there it is.
While you're in there, check "Convert to sRGB", and set Preview to "Use Embedded Profile". This shows you the image with full color management (using the monitor profile to display). Here again SFW uses an outdated default from the old days when browsers weren't color managed.
If you're using a wide gamut monitor some special considerations come in, and in Firefox a manual setting has to be enabled (color management mode 1)*.
As for resizing, I've no particular opinion. SFW seems to do well. But you should do a final sharpening after resizing, which has to be done in Photoshop anyway, so it can probably handle the resizing too. Build it all into a couple of actions and you're set.
*Firefox users should enable this setting in any case, whatever monitor they're using. What it does is to assign sRGB to any and all untagged material. This enables the full color management chain, instead of just passing it straight to the monitor unmanaged. IOW, full color management even on untagged material, as long as it is created in sRGB with sRGB numbers.
