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The visible folders that you see in Thunderbird are controlled by 'subscriptions'.
You can enable or disable the subscription feature. If it is disabled, you will see all folders, including normally-hidden system folders.
You can enable or disable the subscription feature within Thunderbird by clicking on:
Tools → Account Settings → Server Settings → Advanced → Show Only Subscribed Folders
Once the subscription feature is enabled, you can manage the 'visible' or subscribed folders by clicking on:
File → Subscribe
from within Thunderbird. Select all IMAP folders that you want to be visible.
Deleting messages with an IMAP mail client is a little different than deleting messages in a POP mail client. Most POP mail clients will actually move deleted messages into another mail folder. While this is convenient and fast with a local mail store located on your own PC, it is another matter with a server-based message store.
To change how deleted mail is handled, go to:
Tools → Account Settings → Server Settings → When I delete a message
If you have a very large mail store, or if your server is older or slower or if you just like speed, you may opt to choose 'mark as deleted'.
If you like to keep your deleted mail for a while and you have the disk space, you can use the (default) “Move it to the Trash Folder” option.
Other folders that might need adjustments if you use more than one IMAP client might include:
Most users will make use of at least Sent and Drafts.
To adjust the folders into which these items are stored:
Tools → Account Settings → Copies & Folders