Why would Preview save just the annotations

Probably this is not possible with Preview, but there is another (free) PDF reader that allows you to do this (at least your first three points, don't know about converting into another annotation kind): Skim.

After you've opened your PDF in Skim, you can choose Convert Notes... in the File menu. Then you have all your notes from Goodreader in the drawer panel on the right.

This seems to be an interesting problem. I started annotating PDFs on my iPad using GoodReader. Such annotations always show up fine in another PDF readers on other platforms. Then, more recently I wanted to do a similar things on my MacBookPro. At the time, I didn't realize Preview provides this feature. The App Store provides an abundance of choices for PDF annotation programs. I chose PDFProfessional since it only costed a few bucks. I soon noticed it had a problem similar to the one you described. I could see lines and scribble, but not text notes in other viewers unless I clicked and expanded the text box. Otherwise the text box seemed empty.

Last month PDFProessional released an update and this fixed that problem.

It may be be that providing annotations that show up universally is harder than it seems and Preview just hasn't reached that level yet.