Thanks so much for your help. Here's the information you asked for.
The shortcuts work but the icons on the taskbar to the left of the notification area are missing. I can pin a shortcut to the taskbar and it works immediately but doesn't remain on a reboot (though the shortcut still works on my start menu, with the keyboard shortcut I assigned to it and a desktop icon). The shortcuts are also in this folder. The ones I pinned after the problem started are in there too with the number 2 in parentheses after their names.
C:\Users\(user-name)\AppData\Roaming\Microsoft\Internet Explorer\Quick Launch\User Pinned\TaskBar
A few others are in: C:\Users\(user-name)\AppData\Roaming\Microsoft\Internet Explorer\Quick Launch\
(Some are in bothy ...\taskbar and ....\quick launch)
C:\Users\<username>\AppData\Local\Microsoft\Windows\Explorer contains:
ExplorerStartupLog_RunOnce.etl, thumbcache_96.db, thumbcache_256.db, thumbcache_1024.db,
thumbcache_32.db, thumbcache_idx.db, thumbcache_sr.db, ExplorerStartupLog.etl
The most recent of the thumbcache files, thumbcache_32.db, is from April 2017.
The taskband registry key contains: Favorites (reg_binary), FavoritesChanges (reg_dword), FavoritesRemovedChanges (reg_dword), FavoritiesVersion (reg_dword).
Can I do anything with any of this stuff or go with a restore of the windows folder?