Important:

• Since disconnecting each device, I have also modified the Bookmarks on each device.
• I want my laptops to all have an identical set of Bookmarks.

You can look at the replies here:

• [/questions/1151901] Can bookmark names or folder structure be altered after syncing?

Thank you! I will try this and get back to you. Hoping we can create a summary for posterity.

I am disappointed that the official documentation (as opposed to forum entries) is not easier to find on this topic.

I (original poster) have successfully re-connected all devices to Sync on date 2017-01-06 without any major errors by using the following method:

This method is intended for multi-device re-connection to Sync after recovery from a Library or Sync error where Sync is used. If you wish to attempt this process, then please also read the notes posted after the end of this list before you take action!

1. Backup any data (Bookmarks, Add-On data, etc.) that you want to keep safe, via export to a safe folder.
2. Sign in to Firefox account without initiating (signing in to) Sync. https://accounts.firefox.com/signin
3. Change the password. Firefox has a security feature that deletes your entire Library (History, Tags and Bookmarks for all devices) from the Firefox Sync servers whenever you change your Firefox Sync account password (this is not the same account as your Forum account). This is helpful because it allows you to create a clean new Library on the Firefox Sync server, using the [now fixed by you] Library that remains on your device(s).
4. Sign in to Sync on the first computer. Options >> Sync. Alternatively: Menu (3 bars in top right corner) >> Sign in to Sync.
5. Wait at least 1 hour to make sure everything syncs fully. Make sure the device stays connected to Internet during entire Sync reconnect period. If you have already connected another PC, then make sure all connected PCs stay connected during this period.
6. Check the Library to make sure everything is correct, without duplicates and without displaying multiple instances of a single set of Bookmarks/History/Tabs (these are 2 different issues). If you have connected more than one PC, then check every Library.
7. Connect the next computer to Sync by following the steps used for the first PC. Remember to check the Library on every device that has been connected thus far. After all PCs have been successfully connected, move to the next step.
8. Connect the first mobile device (phone, tablet) to Sync.
9. Wait at least 1 hour to make sure everything syncs fully. Make sure all devices connected to Sync thus far stay connected to Internet during entire Sync reconnect period for the current device.
10. Check every Library to make sure everything is correct, without duplicates and without displaying multiple instances of a single set of Bookmarks/History/Tabs (these are 2 different issues).
11. Connect the next mobile device to Sync by following the steps used for the first mobile device. Remember to check the Library on every device that has been connected thus far.
12. Be thankful for the efforts and expertise of the Mozilla group! Could you really code an entire browser like this yourself??

Notes:

• Be sure to also check your Add-Ons and Extensions, as some of these also propagate through Sync.
• Background. The method above was originally used in context of [recovering from a corrupted places.sqlite file] in Firefox's AppData "Profiles" folder (had to delete the file). During the recovery process, all devices were intentionally disconnected from Sync. After the automatic regeneration of a good places.sqlite file upon re-opening Firefox, the user wanted a method by which to spot any errors during each device's re-connection to Sync, one device at a time. The method posted above should not be necessary under normal conditions (i.e. this method is only useful if you are reconnecting due to some error in Firefox). Under normal conditions, simply use normal sign-in to add whatever devices you wish to Sync.
• Minor re-connection error. After re-connecting my first Android mobile device, Firefox on that device told me that I needed to re-verify by re-sending a verification email. I assumed this was due to a bug within Sync's system of connecting mobile devices to Sync (failed propagation, improper timeout duration, etc.). ...so I just waited another 5 minutes. Looks like I was correct. Firefox on my Android went ahead and synced after a few minutes. Also, I could see all of my devices by going into my account on one of my PCs via the URL in step #2. Everything works correctly now.
• Duration of re-connection Sync. Forum moderator "the-edmeister" recommends to wait 24 hours between each device connection, to be safe. Original Poster says: I do not have that kind of patience. For the record, each of my devices synced fully within about 10 minutes, but this is probably because there was not much data to pass because I was reconnecting Libraries which had previously been in Sync.
• Slow computers. One of my laptops is old and under-powered. During re-connection to Sync, Firefox consumed the CPU (100%) for almost a minute. If you have an old/under-powered computer, then expect it to seem to freeze up on you for a bit during re-connection to Sync.

Ok, I have finished the re-connection process. Sync and Library now work correctly on all devices, as far as I can tell.

Moderators and experts: Please feel free to chime in with any ideas regarding improvement or potentially helpful notes for posterity. (Or if I made any typos haha.)