How well is it integrated?
I wouldn't say that it is "integrated" in any meaningful way. WP is a server-side CMS. jAlbum is a client-side album producer. The best you can do is link from one to the other. But you're never going to see, for example, a Tiger or Pluto album magically embedded in your WP site, complete with your WP banner, navigation, etc.
With the bridge, you'll see a generic slideshow of the images, which, when clicked, takes you to the jAlbum album. I suspect most users just provide a simple image or button link from the WP site to their album (ETA: This is what
FabioM has done.). The album itself can provide navigation that takes the visitor back to the WP page.
For example, will a search done in WordPress find an image caption in a JAlbum slide?
No. The album just isn't part of WP. WP doesn't know what it contains.
Can I just upload a new version of a JAlbum album hierarchy and have WordPress pick up the new info?
You can upload a new version of the jAlbum album, and the WP link will still point to it. But WP doesn't know a thing about what's in it.
Are JAbum images and their captions added to the WP media library?
No. Again, jAlbum is
not a server-side CMS. It doesn't do any server-side processing at all.
ETA: The jAlbum Bridge appears to be abandonware. I wouldn't count on seeing a new version that will work with the newer WP block editor, for example.