Neither "spam" nor "unwanted" is a standard Thunderbird folder. I get the impression (correct me if I'm wrong) that you have consciously set up either of these folders yourself.

Spam. If it's really called "Spam" (and not "Junk") then it was put there by someone else; you, your av/security software or your email provider. Whilst Thunderbird can be set to use a "Spam" folder in place of "Junk", it's not likely to have happened by accident.

Unwanted. Again, this is not a regular Thunderbird folder and so we should ask who put it there and who manages what goes into it. Since Thunderbird itself won't set up this folder, (but maybe you did?) any automated movement of messages into it must be managed by either a filter or by some external agency.

The fact that you have these irregular folders suggests strongly that both accounts use IMAP, not POP. And if so, that means the folders in Thunderbird are reflecting folders on the servers.

In both cases, I think the respective mail providers are more likely to be behind this.

"When I down Thunderbird no messages will be moved around until I open it again." So now I'musing webmail to access both accounts. NO message is being moved around. It'd be very strange that two, completely unrelated servers, under two separate admin do almost the same.

Webmail doesn't necessarily use the same set of folders, or the same folder names as we see via IMAP. "Unwanted" could be a feature to filter out message that you might not want to waste time and bandwidth downloading to a portable device.