It may take a while to download all message bodies. Give it some time. You can watch the progress via Activity Manager. Tools (Alt-T) - Activity Manager

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Time to download is directly dependent on the level of interference offered by the ever present and usually painstakingly slow anti virus.

The server sounds like it is disconnecting (that is usually an indication of laggy or non existent communication) Both things that mail scanning is excellent at providing. Then there is the inevitable spam scanner that slows things down to a crawl and to add insult to injury (or that is how I see it) the AV product usually then scans the file that has been opened for updating and slows things down further.

So I suggest you disable any email scanning in your anti virus program. Provide an exception from scanning for Thunderbird profile folder and make sure any other background tasks like download managers steaming sites etc are turned off while the initial update occurs.

Also for the time being set the download frequency for something like 20 or 30 minutes. Outlook.com is very sensitive to login frequency and simply drops connections that occur within their set time frame of 15 minutes. as your expecting the connection to go for a long time a long checking frequency is appropriate.

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"2. Have I correctly executed my wish to have all messages stored locally by having done the following two things: I de-selected “Show only subscribed folders” and selected “Keep messages for this account on this computer”?

No! Make sure everything you do manage to download is moved to a "real" folder, such as in your Thunderbird's Local Folders. Despite the setting to "keep messages on this computer", the storage it enables is intended only as a temporary cache and can't be relied on for persistent storage. Moving or copying to Local Folders ensures you have a full copy on your own hard disk; messages in an account that uses IMAP are effectively remote-view images of messages on a distant server, and may be somewhat transient.

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No, it's just supposed to signpost a useful answer to help others in the same boat.

You have a tricky problem. We don't expect to have to work at an email client to make it get email; that is after all its job. So it should just calmly and quietly get on with it, unseen and unnoticed. When you specifically want it to set about a big job like this, it is frustrating that there's no obvious "get everything" button or switch. And even if there were one, it's quite possible that a busy server would ration your time and your demands on its resources, since no doubt it has some tens or hundreds of thousands of other user accounts to manage as well.

I wish there was a simple answer. I think I'd also go and look at the web portal for the email service to see if it offered any way to export stored messages to a file or files, bearing in mind that what I get at the end might not be easily integrated into my chosen email solution. But at least a private copy of your email, even if tiresome to access, might be better than none at all.

And to attempt to reply to your later message...

If the messages have already been downloaded to the cache, then the copy/move to Local Folders should need only be a local activity, from one place on your hard disk to another. The $64k question is whether all of your messages have bee downloaded already. If not, then we'd fully expect Thunderbird to go and get them, but with all the caveats I hinted at above; servers may only give you so much time and bandwidth at any instant.

You should be able to drag and drop folders between accounts, so I am told. Drag and drop can be a bit dirty; accidents happen and dragged folders get dropped into unexpected places. I prefer a multi-select, then right-click and "move to" for doing this, but this more robust method moves messages but not folders.

If you move messages, the original will be deleted, so you end up with a single copy. If you copy, naturally, you do indeed end up with two copies, the original and the new one. I often advise moving (not copying) messages to Local Folders as a way to free up space on a congested IMAP server.

Mozbackup will certainly give you a snapshot of what you have. If you use it carefully, with Thunderbird offline, then you can move or copy the images of messages from the IMAP account folders to Local Folders, without them needing to also be intact or available on the server. This is handy for users who have switched providers and unwittingly left their mail store on the previous provider's server. We can at least rescue from the cached IMAP account folders.

BTW, you don't need Mozbackup to make a copy of your profile; the profile is just a bunch of files and folders and can be manipulated like any other files or folders in your filesystem.

Mozbackup is valuable in that it hides some of the grubby under-the-hood stuff about pathnames and so makes the backup process more accessible to "ordinary" (i.e. non-geek!) users.

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If you do want both, then copy, don't move.

I use Message Filters in Thunderbird to automate this.

Some people hate "the cloud", don't want google anywhere near their correspondence, want everything local and locked down.

Many of us do want to run email via multiple devices, we don't want to have to worry about what we were using when looking for a message or reply. We use IMAP and leave it all on the server. In fact, I deliberately add messages from other accounts to my gmail and gmx IMAP-based accounts specifically to make use of these accounts' ubiquity.

OTOH I have one particular account which I need to access when out of the office and when I may not be able to connect to the server at my workplace, so for this I use my Local Folders "backup" of the account.

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Your Local Folders is part of your profile.

https://support.mozilla.org/en-US/kb/profiles-tb

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Download and install Thunderbird on the new computer but don't add any accounts. Close it and you will find it it has created a new empty profile. Copy the profile from your old machine over the new one and it will look just like your current Thunderbird, complete with messages in Local Folders.

Some of my "local" folders are not staying synchronized. For some of the folders that I frequently move messages to from my InBox, even some messages today (3/21/2015), when I open the folders, the latest email, for example, are dated 3/8, 3/11, 3/15, 3/6, etc.

When I look at my email on the server (Yahoo / AT&T), the messages moved since those dates are there. If I move one of those messages back to the InBox, that message will then appear in the Thunderbird InBox.

It is as though the folder is full?

As far as I can tell, all my Settings are correct and match those mentioned in the thread above.

One problem I may have - all my folders on Thurderbird are in the IMAP account. See the attached screen shot. Should those folders be moved to Local folders, which shows up as an account below the IMAP account in the folder listing? The only two folders there are Trash and Outbox. See the 2nd screen shot.

I am running Windows 7 professional and the latest version of Thunderbird - 31.5.0.

The Local Folders account in Thunderbird will not help you. It does not synchronise to servers. It represents messages that are stored on a local device (usually this means your hard disk). It can be useful when you specifically want to store a message separate from and independent of an IMAP-connected account.

Why do you have both IMAP and POP set up on the same email account? This isn't a common configuration and may in itself cause difficulties.