Do they say exactly what is suspicious about the emails ?
Set the Internet Mail provider to High sensitivity, this should then recognise if you are sending out lots of identical email (spam bot).
What is your firewall, as this should also prevent unauthorised outbound connections ?
If you haven't already got this software (freeware), download, install, update and run it, preferably in safe mode and report the findings (it should product a log file).
1.
SUPERantispyware On-Demand only in free version.
2. MalwareBytes Anti-Malware freeware version
http://download.bleepingcomputer.com/malwarebytes/mbam-setup.exe, right click on the link and select Save As or Save File (As depending on your browser), save it to a location where you can find it easily later. Also Try this tool, RogueRemover, available here
http://www.malwarebytes.org/rogueremover.php