When you configure the SMTP server to use, there is an option to 'test'. This will send a test mail. You don't recieve this either? If not I think first of all you should check firewall settings for port 25 on both the mahcine where SQLyog/SJA is running and on the machine/network where the SMTP server is running. You may try temporarily to tun off what firewall and firewall-like programs you have installed.
Also: Can you use a traditional mail client (not webmail, gmail or whatever of the kind) but programs like Windows Mail or Outlook on that SMAPT server? There should not be any difference between sending a mail from Outlook etc. and from SJA. Settings details are exactly the same.