1) As of right now the addon does not let you search through all the bookmarks. Only those that you save or add inside the (Bookmark Menu -> SessionSync) folder. If you look in the bookmark manager you can see the folder there. By Advanced Bookmark Manager I meant that the addon can be used with that purpose in mind because you can manually save every tab, group of tabs, order them, edit, delete, etc but only if you do this through the addon UI.
2) Automatically saved sessions (those from History Tab) are not saved in bookmark because that might cause performance issues. Saving is done manually when you need to save something. I would say that the addon fits perfectly somewhere in the middle of an Bookmark Manager and a Session Manager, because it's mostly intended to have good control on what you want to do, and offers a very common Bookmark Manager experience, as well as because all the saving is done in Bookmarks.
NO, as long as you have Firefox Sync enabled (or whatever bookmark syncing solution you might use) all your bookmarks will be synced. Like I mentioned, all your Bookmarks are saved in the [Bookmark Menu -> SessionSync] folder so of course you will be able to find them there and interact with them even if you don't have the addon installed. For example that's what I do on my mobile phone. I access them through the Firefox mobile boomark manager because I don't a mobile version yet, but I can still access all my saves very easy.
3) In the addon panel you have a search bar in top-left. Or simply after you open the addon panel, press CTRL + F and then start typing. Search only works for folder names at this moment (sadly). If you want to search through Bookmarks... for the moment the Firefox URL bar can do that ... yes you will search through all the bookmarks not only those saved by the addon but at least it works.
In case you didn't seen the Tutorial, please look over it because it might explain a lot of the addon features (with images also). The tutorial can be access from the addon menu (top-right button from the addon-panel)
If you have other questions regarding the functionality you can also contact me over the mail. Just press on the "Send Feedback" button from the addon menu and the mail application should open with my address already completed.
Thanks for feedback