What are the black discs? Do they have a name?
They are just called Washer and Ring fittings. Metal or plastic washer but with a slit instead of a round hole.

Are the rings hidden in the lining of the jacket so they wouldn't touch your skin?
Yes they are inside the jacket lining. The ones on the photo are on the pocket flap. The lining was not sewn at the top to allow access to the buttons. There should be somewhere to get under the lining without cutting. You may have to undo a few stitches then sew up again.

There is a thread here on this same subject.
http://www.xmarksthescot.com/forum/f...buttons-66320/

Chris.