with all three using IMAP will the same folders and emails including sent items and replies appear on all three computers?

Yes, that's the whole point using IMAP.

Are these the correct settings for true synchronization?

That should do it. But you may also check the synchronization status for individual folders. A good test is to switch Thunderbird into offline mode and then open a message. It should open (the local offline copy) without a hitch.

Having said that, using IMAP isn't a replacement for creating backups of your Thunderbird profile. Remember, deleting a message on one device means it's gonna be deleted also on the server and all other devices accessing the account via IMAP.

I thought this was solved but there's a glitch. I was on my laptop the whole day and checked mail, wrote responses and all that. I just got home and opened Tbird on my desktop but there were emails that I had left on the laptop that did not appear in the desktop Tbird. I had to use the ajsystems back up program to back up from the laptop to the desktop to get those missing emails showing. I thought that in using IMAP whatever emails I had in my inbox on one machine would appear on the other machines that use IMAP. But this was not the case. I'm sure I'm missing something but have no idea yet what that might be.

Can you post the Troubleshooting Information from the two computers? Help (Alt-H) - Troubleshooting Information

WARNING:

 This contains sensitive information which shouldn't be forwarded or published without permission.

 Application Basics

   Name: Thunderbird
   Version: 38.1.0
   User Agent: Mozilla/5.0 (Windows; U; Windows NT 5.1; en-US; rv:1.8.0.9) Gecko/20061207 Thunderbird/1.5.0.9 Mnenhy/0.7.4.0
   Profile Folder: C:\Users\Philip\AppData\Roaming\Thunderbird\Profiles\snjybqrk.default
             (Local drive)
   Application Build ID: 20150707103124
   Enabled Plugins: about:plugins
   Build Configuration: about:buildconfig
   Memory Use: about:memory

 Mail and News Accounts
   account1:
     INCOMING: account1, Local Folders, (none) Local Folders, plain, passwordCleartext

   account4:
     INCOMING: account4, [email protected], (imap) imap.earthlink.net:143, plain, passwordEncrypted
     OUTGOING: smtpauth.earthlink.net:587, plain, passwordCleartext, true

 Crash Reports

 Extensions
   CompactHeader, 2.1.0, true, {58D4392A-842E-11DE-B51A-C7B855D89593}
   Hide Local Folders, 1.04, true, [email protected]
   JunQuilla, 1.0.4, true, [email protected]
   Mark All Read Button, 0.7.1, true, {DA506923-55C4-4fbe-9E4C-D24B80D7DE9F}
   Bitdefender Antispam Toolbar, 7.0, false, [email protected]
   Cloudmark DesktopOne, 1.7.0.27, false, [email protected]
   Lightning, 4.0.1.2, false, {e2fda1a4-762b-4020-b5ad-a41df1933103}
   Signature Switch, 1.6.13, false, {2ab1b709-ba03-4361-abf9-c50b964ff75d}

 Important Modified Preferences

   Name: Value

     accessibility.typeaheadfind.flashBar: 0
     browser.cache.disk.capacity: 358400
     browser.cache.disk.smart_size_cached_value: 358400
     browser.cache.disk.smart_size.first_run: false
     browser.cache.disk.smart_size.use_old_max: false
     browser.display.foreground_color: #330033
     browser.display.use_document_fonts: 0
     extensions.lastAppVersion: 38.1.0
     font.internaluseonly.changed: false
     font.minimum-size.x-western: 16
     font.name.monospace.el: Consolas
     font.name.monospace.tr: Consolas
     font.name.monospace.x-baltic: Consolas
     font.name.monospace.x-central-euro: Consolas
     font.name.monospace.x-cyrillic: Consolas
     font.name.monospace.x-unicode: Consolas
     font.name.monospace.x-western: Arial
     font.name.sans-serif.el: Calibri
     font.name.sans-serif.tr: Calibri
     font.name.sans-serif.x-baltic: Calibri
     font.name.sans-serif.x-central-euro: Calibri
     font.name.sans-serif.x-cyrillic: Calibri
     font.name.sans-serif.x-unicode: Calibri
     font.name.serif.el: Cambria
     font.name.serif.tr: Cambria
     font.name.serif.x-baltic: Cambria
     font.name.serif.x-central-euro: Cambria
     font.name.serif.x-cyrillic: Cambria
     font.name.serif.x-unicode: Cambria
     font.name.serif.x-western: Arial
     font.size.fixed.el: 14
     font.size.fixed.tr: 14
     font.size.fixed.x-baltic: 14
     font.size.fixed.x-central-euro: 14
     font.size.fixed.x-cyrillic: 14
     font.size.fixed.x-unicode: 14
     font.size.fixed.x-western: 16
     font.size.variable.el: 17
     font.size.variable.tr: 17
     font.size.variable.x-baltic: 17
     font.size.variable.x-central-euro: 17
     font.size.variable.x-cyrillic: 17
     font.size.variable.x-unicode: 17
     general.useragent.override: Mozilla/5.0 (Windows; U; Windows NT 5.1; en-US; rv:1.8.0.9) Gecko/20061207 Thunderbird/1.5.0.9 Mnenhy/0.7.4.0
     gfx.direct3d.last_used_feature_level_idx: 0
     mail.openMessageBehavior.version: 1
     mail.winsearch.firstRunDone: true
     mailnews.database.global.datastore.id: a98b8a46-03d1-4ff9-9106-4ef02b175c0
     mailnews.database.global.views.conversation.columns: {"threadCol":{"visible":true,"ordinal":"1"},"flaggedCol":{"visible":true,"ordinal":"3"},"attachmentCol":{"visible":false…
     mailnews.database.global.views.global.columns: {"threadCol":{"visible":true,"ordinal":"1"},"flaggedCol":{"visible":true,"ordinal":"3"},"attachmentCol":{"visible":false…
     network.cookie.prefsMigrated: true
     network.predictor.cleaned-up: true
     network.protocol-handler.warn-external.blocked: false
     places.database.lastMaintenance: 1439293391
     places.history.expiration.transient_current_max_pages: 104858
     places.history.expiration.transient_optimal_database_size: 21057536
     plugin.disable_full_page_plugin_for_types: video/x-ms-wmv,audio/mp4,audio/mpeg,audio/x-m4a,image/tiff,audio/wav,video/quicktime,application/pdf
     plugin.importedState: true

account4: INCOMING: account4, <snipped>@earthlink.net, (imap) imap.earthlink.net:143, plain, passwordEncrypted

Can you confirm 'passwordEncrypted' is correct? I don't think it is. Try 'Normal password' instead.