I'm not sure why, but every time I want to check for new messages in my Gmail account, hundreds of messages from 2006-2009 appear. I want to delete them from my account without downloading them, but I haven't been able to do so yet.
Will I have to download them all and let Gmail delete them from my Gmail space? Is there no other way?

You have to configure your Gmail POP3 settings properly. Look the image below where you can see the default POP3 setting: archive Gmail's copy. With this setting, every time you command Gmail to delete a message it will archive it instead.Choose the proper setting to fit your needs and go on.

Concerning old messages that still on server, open All Mail folder (use the Web interface), select and delete them. Maybe a filter will be the best way to select the whole bunch.
Good luck!

Thank you!