Download messages from other folders
IMAP accounts don't support the concept of downloading since they have a client server view of the world. You can copy/move messages/folders in either direction from local to remote folders. POP accounts are download-centric. The only way to access messages using the POP server is to download new messages from the inbox folder by checking for new mail.
However, if you use the webmail extension or a webmail add-on such as Ypops! you can configure it to download messages from other folders and include the messages in the inbox folder. It can work around that limitation because it uses screen scraping to fetch the message instead of the POP3 protocol.
YahooPOPs! adds an email header called "X-YahooPOPs-Folder:" , FreePOPs! adds "X-FreePOPs-Folder:" and the Webmail extension adds "X-Folder:" to specify the folder the message was downloaded from. You could create a message filter in Thunderbird to move that message to the desired folder based on the headers contents.
Frequently there is a timeout problem the first time you use the webmail extension or a add-on to fetch messages, due to the large number of messages. You may need to increase the connection timeout or see if the extension/add-on supports some sort of limit option. For example, with Ypops! you can right click on the icon in the system tray, select configure, receiving mail, and then set the "maximum number of messages to download per pass".