Your album is already a set of HTML pages and yes, you can publish to your own website, but you don't need to "embed" it to another HTML page. Why not just have a link to it from one of your existing pages?

First, yes, you can host your albums on any webhost you like. The albums themselves are just HTML, CSS, and Javascript - there is nothing about them that ties them to the jalbum.net site.

But jAlbum does not produce "HTML snippets." It produces complete web pages - in fact, it can produce complete websites. They're not designed to be embedded into other pages. If you wanted to show the CNN website, for example, would you try to embed it into your own site? Possible, but it would work very badly.

jAlbum albums should be allowed to stand on their own. You can link to them from your own pages, and the jAlbum album can have a simple link back to the calling page.

The other thing you can do is embed your own custom code into your album to do things like providing a custom header and menu system. Here's a simple example, from my skin demo site. The page was produced with one of my skins. But I've told the skin to include a simple menu ribbon at the top. So, the jAlbum page isn't embedded within another page, but it includes extra material - the menu bar - that isn't actually built into the skin:

https://jefftucker.jalbum.net/skins/neptune/

If you're running a Wordpress site, then you can install the "jAlbum Bridge" plugin that allows you to display embedded galleries on your existing pages. When clicking these embedded galleries, you're then take to the "full window" gallery. All at https://www.your-site-here.com/

Not really. Very small jumps. Linking from the album back to your own site, for example, is virtually automatic in any jAlbum skin - no coding required. Planting your own menu system within the album, on the other hand, would require some basic HTML/CSS/Javascript skills.