Another point to remember is that many external mail servers will reject messages coming from mail servers on dynamic IP's, so you need to send mail via your ISP's smtp server, you can set that in the server manager Email panel eg mail.isp.com. You can also set smtp authentication (if needed) in the same area, as per previous posters suggestion.
Correct. I had forgotten this was added in SME 7. You can tell how long ago I may have done it.
In general it seems email is working on the server and you have domain redirection working.
I agree with Mary that you should double check what she identified (including the MX record at your domain hoster).
But given neither inbound nor outbound mail is working from your PC client other than when it is to someone else on the server, it seems possible something else on your LAN or ISP is intercepting this.
What happens when your PC client tries to retrieve email directly from an external POP? What happens if the PC client tries to send directly via your ISPs SMTP server?
Do you have any other devices besides your client PCs, SME server, and Router?
What type of router is it? Does it understand email as an application? If something else understands email it may be intercepting it. Your port forwarding would affect inbound but not outbound email. But outbound email can be intercepted.
Is SME run as "server only".
Is your router IP address set as the default gateway for your SME?
If you are sure you have everything set properly and double checked, then you will need to isolate the issue. Determine if email is getting to your router, if it is being received at all by SME. You will find it but the trick is to figure out where the messages are being lost.