I keep getting email from webmaster@mydomain.com / support@mydomain.com and so on. It has to be some type of spam and I dont even have email addresses set up for them. Is there any way to keep my server from sending these?

Check the /etc/aliases file, also check sendmail's virtusertable file. Remove those addresses if they exist, or you might have a catch-all enabled (which you obviously would need to disable also)...

It is probably that virus/worm which pretends to be from your sysadmin. Just delete them.