Windows Support Scam - I was called by someone who said they were windows support and my computer was sending them warnings and messages!

Today, while composing a gmail suddenly my curser became a tool to black out my emails and everything in the gmail box/window.  When I moved my cursor outside of the gmail box then the cursor stopped blotching everything in black.  Naturally, like all good very amateur computer users, soonafter I rebooted my computer...but the problem still continued a little bit then stopped.   I have Norton antivirus program...the premier edition but it did not detect/warn me of this problem.

 So, then about 15 mins later, I get a phone call (caller ID displayed a name starting with Jeffrey... he probably stole this name from some innocent person)  he requested to talk to the 'owner' of my phone line. He was annoyed and talked to me like I was stupid when I asked questions about who he was and how did I know if he was legit.  He then made the same pitch/comments that others have noted above..."I am from Windows Operating System and we have detected malware that is duplicating rapidly and is doing damage and will spread".   When I started asking him questions the guy became more impatient/irriated!  Another clue something was wrong.  Then I asked him for his name, his company phone, City and State...he said in his thickly accented, I think Asian, voice I am Daniel Higgins (for the real Daniel Higgins....I think your name was hijacked for this scam). I asked him for his office number, city and state. He then gave me a 800 # , said he was in California..and he mumbled an undiscernable city with much annoyance.  Later, I did call the 800# and learned, of course, that there was no such Daniel Higgins there, and that there was no IT people who fix computers there. In fact the guy who answer seemed a bit grumpy and got to the point right away and quickly hung up. 

Later, after speaking to our office IT person it was suggested to run my antivirus- anti-malware programs and then restart the computer.  This thoughtful IT person also  provided a Microsoft Security Scanner program link to download & run....which I've done....and which the final report indicated  there were no viruses or malware found.   

I was concerned how this guy not only accessed my computer but also,  got my phone number after going on my computer.  The IT guy from work said they look for vulnerabilities on the computer and find an IP address then are able to trace it to a home phone.  Thankfully, all the advice from those who educate the public about warnings signs of scams came to the forefront.    But I still wonder, since they accessed my computer once...will/can they do it again! What other steps can I take?  AND has he planted a virus or malware?  Could he read my email? And lastly, why didn't my Norton program detect him....is it because he stayed in the gmail box/window?