Can't delete emails by highlighting and clicking delete in toolbar. The email goes grey and doesn't delete?!

Not certain, but this can fix myriad Mail problems...

Safe Boot from the HD, (holding Shift key down at bootup), it will try to repair your Disk Directory while the spinning radian is happening, so let it go, run Disk Utility in Applications>Utilities, then highlight your drive, click on Repair Permissions, then move this folder & file to the Desktop.

Move this Folder to the Desktop...

/Users/YourUserName/Library/Caches/Mail/

Move this file to the Desktop...

/Users/YourUserName/Library/Mail/Envelope Index

Reboot.

If that doesn't do it and you can afford to redo all your Rules, try these & reboot...

/Users/YourUserName/Library/Mail/MessageRules.plist

/Users/YourUserName/Library/Mail/MessageRules.plist.backup

Note, in 10.5 & up /Users/YourUserName/Library/Caches/Mail/ may or may not exist.

My problem was similar: Whatever messages I deleted from whatever mailbox/server, they always 'came back' after reopening mail. I moved (deleted)

/Users/YourUserName/Library/Caches/com.apple.mail

and that soved my problem. I am running OSX 10.9.2

Thanks for the report! 🙂