How can I link Fields in a View to the entity it belongs to?
The link to node or link to user option you see on some properties in views are special. The reason the possibility is there, is because Views has added the option and functionality in a special handler.

Things are not that easy with fields since they can be attached to anything, which makes it much more complex (compared to nid, uid, title and username). You could probably do this with some code, but usually it's much easier to use the rewrite result option, and select output as link and type in the link with tokens: node/[nid]. I'm not sure, but I believe the Views will run the link through l or url enabling path auto.