New Emails in Trash Folder

It sounds like a spam filter gone awry. You don't say what email program you use, or if you've installed any anti-spam software. But if you've installed a spam blocker or added filtering rules to your email client, that may be the cause.

I must admit that part about incoming emails showing up in the Drafts folder is very odd. That makes me wonder if some sort of malware is involved. Have you done a virus & spyware scan?