Various and sundry attempts to use my own ISP through exim4

[David Baron <d_baron@012.net.il>]

I have a dynamic ISP. I can use it of SSH, etc.

If I try sending email to it using my provider:
mail to root@mydom.ain is "completed" by exim but I never see the message.
mail to user@mydom.ain goes to neverland. Eventually, the provider's system 
sends an error message after four days of timeout.

If I try sending email similarly but using the "mail" commandline:
To my provider-email address name@dom.ain, is rejected saying name.dom.ain 
does not exist!!
To my own ISP--well, the user UID != mail UID so it does not work.

My current exim4 has my provider as the smart smtp server and routers for each 
user@mydom.ain to /var/mail/user. 127.0.0.1 is NOT specified for listening.