Please clarify your goal. For example, is it that you want to download all of the messages to your PC and then delete the account? You indicate wanting to delete the messages, but you also seem concerned that the deletion will be permanent. This needs to be clarified. With specific information, we can assist. Thank you.

Hello David, I reached this point (i.e downloading Thunderbird and linking my gmail account) in order to ultimately free up space on my gmail account by using the power of TBs' filters. I have started filtering all my old emails into folders and am ready to delete them in gmail. However, I want to keep some of the filtered emails and I'm weary that when I delete them from gmail (in order to create more free space) I'll lose them permanently because I may not have completed the set-up properly. Does this explain it sufficiently?

It still is not clear. If this is an IMAP account, the messages are not backed up unless you copy them to a local folder nad have spot checked to ensure you have complete messages, not just headers. Otherwise, the messages on PC are just copies of originals and, if deleted onlline, they're also deleted on PC. If you want to keep messages on PC, and delete from online, I suggest - check account settings>synchronization&storage advanced tab to verify that the folders can be accessed offline - then, click File>offline>download to ensure that all message content is downloaded. - then, create a Local Folder and copy the desired messages for keeping to the Local Folder - then, you can delete the Gmail copies. - at this point, those messages exist only on the PC, so be sure to do occasional backup of profile.