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Hi digitalfringe and welcome.
These things are difficult to diagnose but as it seems to work OK after a factory reset, it sounds like a software issue. Have a check of the following possibilities: -
1. All systems become cluttered after a lot of use. Registry links often remain after software has been uninstalled and your Vaio will take time looking for these links. Try a Registry Cleaner to remove broken links, delete the internet cache and remove unwanted cookies. You can use a free trial of something like System Mechanic or use a freeware program like Ccleaner. This may speed up your Vaio and can be used frequently to prevent a build up..
http://www.iolo.com/default.aspx
http://www.piriform.com/ccleaner
2. What internet security/firewall are you using? Sony preinstall McAfee and if you are using somthing else, there can be a firewall conflict which blocks downloads. Check you have completely removed McAfee using the removal tool - MCPR.exe.
3. Your Vaio may be overheating causing the processor to throttle or shut down to prevent damage. Check your CPU and GPU temperatures using free software like PC Wizard. You can try to remove dust and fluff using a can of compressed air but if it is bad, the Vaio may need taking apart and cleaning internally. Using the Vaio on a soft surface like a bed or carpet or in a dusty environment make this worse.
http://download.cnet.com/PC-Wizard-2012/3000-2094_4-10793737.html
If these don't help, come back and we can try a few more things.