Some companies send out junk mail at random, they don't even know if the addresses actually work. When you click on 'unsubscribe', that tells them that your address does work and that you read your mail -- so that encourages them to send more junk, and to sell your address to other companies as well. So be careful before you unsubscribe. Is the message from a reputable company you know? If so, then by all means tell them you don't want further messages. In such cases the 'unsubscribe' link often takes you to the company's website which gives you various options to manage how often you hear from them: the company that provides my mobile phone service is like that. But if you've never heard of the company, and the text of the email doesn't address you personally (not using your name or quoting your customer number, for example), then the message may be one of those I described at the start. In such cases the best course is not to unsubscribe and not even to open the mail: just delete it.

Unfortunately you have no control over who sends you anything. The server will accept what it is sent providing it meets their standards and there is space in the receiving account.

Thunderbird simply downloads whatever is in the Inbox or if IMAP then any subscribed folders.

I would mark the emails as Junk or immediately delete.