You can make Local Folders in Thunderbird and copy the messages that you want to save into those folders for safe keeping. I don't know what you mean by 'Thunderbird server' - TB is not an Email provider and does not have a server. It is also good housekeeping to back up your Profile at least once a week to safeguard against data loss.

There is no such thing as a "Thunderbird server". I presume you mean you want to store your mail locally on your computer using Thunderbird.

You may create local folders in Thunderbird to store and organise your mail there however you wish, but removing it from Gmail is tricky because Gmail doesn't work like a normal IMAP server would, and one of its many particularities is that Gmail keeps all messages in a special folder called All Mail even when you remove them from other folders (labels in Gmail), and there is no way for mail clients such as Thunderbird to directly remove the messages from there.

The only way to remove a message from Gmail's All Mail with Thunderbird (or any other mail client) is indirectly by moving the message to Bin/Trash first (has to be the same Bin/Trash as the one used by Gmail for this to work), and then remove the message from there or just let Gmail automatically do that after 30 days.

So, in order to accomplish what you want using Thunderbird alone, you'd need to copy the messages you want to keep in Thunderbird to a local folder first, then move the original messages to Bin/Trash in order for them to be eventually removed from Gmail server.

Alternatively, you may simply move the messages where you want to store them in Thunderbird, and every now and then log into your Gmail account using webmail to delete your messages from All Mail there.