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What happens to that message from the original gmail account point of view?"
If you move a message from a Gmail IMAP account folder to a folder in another account the original message is copied to the new location, and then the original deleted. Deleting it sets a flag that hides it from view (unless you change how delete works per "when I delete a message" in the accounts server settings) and marks it as deleted. It is not physically removed until you compact the folder or you expunge the folder (by enabling "clean up (exunge) inbox on exit" in the accounts server settings).
I'm assuming that you didn't enable the setting in gmail webmail to tell it to ignore a email client deleting a message. I'm also assuming there was only one copy of the message, and ignoring the existence of labels and the All Mail folder. See
http://kb.mozillazine.org/Gmail if you want to learn about that.
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What would happen if I moved ISP and the email address was deleted, would I still have those gmail messages? "
No. If you moved a message from Gmail to your ISP account and then lost the ISP account all of your copies are probably gone. If you want to keep the original, use copy, not move.
However, deleting an account does not delete its files. By default Thunderbird creates "offline" folders that mirror the content of the folders in the IMAP account. They are mbox files (a 7bit ASCII text file named after the folder, with no file extension, that contains all of the messages for that folder), just like what POP accounts use to store messages for folders. You could import those folders into "Local Folders" using the ImportExportTools add-on.
The danger of relying upon offline folders for backup is if you delete a message in the corresponding remote folder, it is physically deleted in the offline folder when the two folders sync. When you are working "online" Thunderbird displays the contents of the remote folder. When you are working "offline" (file -> offline -> work offline) it displays the contents of the corresponding offline folder instead, if it exists. The synchronization & storage settings in the account settings manage offline folders.
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And is gmail a unique case or does the same apply to gmx, outlook.com emails, etc?"
Gmail has some unique features that I've glossed over but they're not really relevant. The issues you raise would occur with any IMAP account.
If you move a message to a folder in the Local Folders special account it will still exist even if you delete all of you IMAP accounts. The disadvantage is that its now stored on your hard disk and you need to worry about backing up your profile. You also can't access that mail from another machine.
I have 9 accounts, including ones from yahoo, gmail, gmx, outlook, zoho, my isp etc. Gmail IMHO is the best and most reliable of the free email providers despite its quirks. However, I pay for a inexpensive account with Fastmail.com which I use as my main mail account, and the one that I use to store any messages that I would be upset if I ever lost them. I've found Fastmail to be extremely good about making certain that their backup/recovery mechanisms will prevent you from ever losing a message, do a excellent job of detecting spam, and I like being able to get support from my email provider. I've had an account with them for over a decade. Their character has changed in recent years (they are now more focused towards small business users rather than power users like myself) so I'm now on a legacy plan but for the moment they still seem my best choice.
ISPs treat email as a cost center, they usually do the minimum they can get away with. The problem with free email providers IMHO is that they can lull you into complacency. Things may work great for several years and then when something bad happens you may discover the downside of being a user, not a customer. My current shortlist of alternatives to fastmail is runbox , polaris, posteo.de and mailbox.org. I'd definitely avoid a "hosting" provider.
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Is there a better way to collect messages about a topic from multiple email accounts than using TB folders?"
Let me think about that for a bit. How many topics we're talking about has an impact.
