E.g. I did a mograph piece for a local Jazz Festival. It's okay to include this in my portfolio and/or reel, right?
For sure if it's aired already. In fact I'm sure a local festival is happy for any extra publicity, any extra people seeing the piece.
I did many pieces for a New York company who is licensed to stream Korean TV dramas online. The work I did includes clips from the shows, music from the shows, etc. That company is fine with me including it in my portfolio. But could the Korean networks give me trouble?
That's not your problem. If the people you worked for cleared you to use it then you can use it. On a practical level what you are doing is probably pretty far off the radar for a Korean network.
My rule of thumb is if you are doing something that is going to be publicly broadcast (on TV, in theaters, or put on the web) then once it has aired you should be able to use it in your reel as long as there is nothing in your contract that states otherwise. To be on the safe side I put a clause in my contract that says I can use any works produced under the contract for my promotion as long as it has aired. No one usually has a problem with this, sometimes producers ask for a watermark from their studio or to specify a certain delay after the air date.
If you are doing something that's internal or B2B like a sales video, something for a trade show, an annual meeting, then the opposite assumption applies. I would assume that I'm not allowed to show that unless I specifically ask. A lot of time they will let you use pieces that don't contain any information, like if you did a nice logo animation they'll say use that but nothing that shows any information.