Make sure that toolbars like the "Navigation Toolbar" and the "Bookmarks Toolbar" are visible:

• Firefox > Options
• "View > Toolbars"

Open the Customize window to set which toolbar items to display:

• View > Toolbars > Customize
• Firefox > Options > Toolbar Layout

• Check that the "Bookmarks Toolbar items" is on the Bookmarks Toolbar
• If the "Bookmarks Toolbar items" is not on the Bookmarks Toolbar then drag it back from the toolbar palette in the customize window to the Bookmarks Toolbar

• If missing items are in the toolbar palette then drag them back from the Customize window on the toolbar
• If you do not see an item on a toolbar and in the toolbar palette then click the "Restore Default Set" button to restore the default toolbar set up

• https://support.mozilla.org/kb/How+to+customize+the+toolbar
• https://support.mozilla.org/kb/Back+and+forward+or+other+toolbar+items+are+missing

The only search-box in the tool-bar palette is a Yahoo one, which i do not want.

Did you try the Restore Default Set button?
That should bring back missing toolbar items.

You can check for problems with the localstore.rdf file if this didn't help.

• http://kb.mozillazine.org/Corrupt_localstore.rdf

The Restore Default Set button brings back the Yahoo search box (which I do not want), but not the Firefox search box.. Using the localstore.rdf file seems to be very complicated.

The point is to delete the localstore.rdf file or rename to localstore.rdf.sav to force Firefox to create a new file in case the file is corrupted.
That should bring back missing toolbar items.

I located the localstore-rdf file, closed Firefox, renamed the file, and restarted Firefox. And this restored the Yahoo search box! (but not the Firefox search box). But I have decided I can live with it, as the "Add to Search Bar", and "Search Bar Autosizer" add-ons work OK.