When you shoot RAW + JPEG the Photos app stacks the two images together. The J means you have a stack with a JPEG file and a RAW file and the JPEG file is used as original. To change to the RAW file go into edit mode (enter) and choose Image - > Use Raw as original. The letter will now change to a R.
It means that you cannot change the volume for the selected output device. For example, if you connect your mac to a TV using a HDMI cord, you will not be able to control the TV's volume with your mac. You'll have to use the TV's remote instead.
This shouldn't ever happen when you connect something into the headphone jack though. Perhaps you have a faulty set of headphones?
If you haven't already, turn on the sidebar - Option-Command-S - then look towards the top and under Shared, click Activity. Likely there is some activity on a shared album or Photo Stream. It may also be someone inviting you to join a shared album or Photo Stream. (I'm not sure how that would show up in Photos, as I haven't had any invitations since Photos was released.)
Either way I'm guessing once you take a look at that area, the badge will disappear.
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