Many thanks to those of you who offered suggestions. I checked the antivirus, and I was moments away from doing another uninstall, when I got the urge to check my firewall settings. I set new ones for Skype, rebooted and prayed. To my utter shock, that turned out to be the issue! Skype was having difficulties in getting through the firewall. Now that it has proper firewall protocols, it's running exactly like it's supposed to.

Thanks to everyone who helped. :D