As a general rule, save only your master digital files, export JPGs for sending to the lab or uploading to the web, and then trash these derivative files so as not to have multiple versions of a file that will need to be cataloged.
From Photoshop, use the SAVE FOR WEB feature and click the sRGB box. This works well, but is my second choice.
From Bridge, use TOOLS--PHOTOSHOP--Image Processor. This is my least favorite way to handle this. Not many options available here.
Best is from a Lightroom catalog, using an export preset. Much easier to catalog and find your master files as needed, and then export your derivatives with full color space conversion, sharpening options, resizing options, file formats, etc.
If you don't have Lightroom, it is well worth the small investment. Bridge is okay but is not a robust cataloging app, and if your images aren't cataloged, you're wasting precious time.
I would not bet $0.50 that any Windows media product respects embedded color spaces, so I would not trust it. Export from an Adobe product, upload and see how the images look on several different monitors.
Feel free to peep me off line if you need more help.