A full URL, including the https:// (or http://) part, for example https://example.com/my-page/. Anything else, including a bare domain like example.com or a path like /my-page/, is not saved.
No. Since 1.6.1 only full URLs are accepted, so a link always means exactly what it says. To link to another page on your own site, enter its full URL. If you move your site to a new domain, a search-and-replace of the old domain covers these links along with the rest of your content.
Clear the field in the Page Links To panel and update the post. The link is deleted and the post goes back to its normal URL.
Yes. By default it works with posts and pages. Use the lh_page_links_to_get_applicable_post_types filter to change the list; see Hooks and filters. The plugin is about decisions, not options, so there are no settings screens, but there is likely a filter for most behaviour, and if there is not, one can be added.
LocalHero plugins are made to WordPress standards, so they should work with well-coded plugins and themes. If something does not work, first deactivate all other plugins and switch to a default theme (such as Twenty Twenty-Five) to see whether the problem persists. If it does, please post in the support forum.
Contact Peter Shaw for custom work and enhancements at shawfactor.com/contact.