As Scott says, you must ensure that the web version of your image is created in sRGB, because that is the color profile used by all web browsers. Any other profile will result in undesirable color shifts.
You should not embed the profile in files destined for the web, because it adds file size without adding anything to the rendering of the image in a browser.
You should embed profiles in any files destined for print, with the exception of PDFs (which are best exported as PDF/X-1a to ensure there are no production problems). Spot colors from the Pantone or other color libraries are what they are, regardless of color profile, but it doesn't sound as if you're using spot colors this workflow.
Your challenge is to minimize any opportunities to alter your artwork, so my overall suggestion would be that you not send Photoshop files all. Use PDF/X-1a for print and 24-bit PNG (converted to sRGB before saving, using either "Perceptual" or "Relative Colorimetric" and black point compensation) for anything that is destined for on-screen viewing, whether on the web or on a digital signage screen. The PNG format doesn't include a color profile, and assumes sRGB as the output intent.