It would be round about now that I ensure everything is backed up, and start reaching for the Windows disk, formatting the drive, and re-installing. It's often just the quickest way to fix arsy little problems like this.
You've clearly had this problem for some time now. Re-installing takes a couple of hours unless you've got a million apps installed, so you have to weigh the time already spend against 2 hours re-installing. Also, even if the HDMI problem persists after re-installing, at least you will know it's a hardware related problem with your GPU and you can stop searching around for a software related fix. It will probably also fix your update problem.
This is one advantage to having a full back up system, and not just a means of backing up your important files, but a complete image of your system drive. If something weird goes wrong at a system level, you can immediately restore from a back up from say... 48 hours ago, and whatever weird ass problem caused it will just be put back as it was. That's even faster too. I can restore from a disk image in 30 minutes.
Acronis True Image has been utterly reliable and flawless for me. A full disk image back up is not just for security, it has the advantage of being able to fix these weird little problems that crop up from time to time if you restore as soon as you notice it. Try and live with it for a week however, and the back ups will start to also contain the problem. I do incremental back ups. Back up 1 is a full back up, then the next 7 back ups are incremental, then the 8th is a full back up and that cycle starts again. As I back C: one night, D: the next (fully automated), I have 14 days in which I can go back in time should a problem arise. If you ensure your images and files are on a separate physical disk from C: then all you're doing is restoring your system, not your images. Far too many people only back up their files, not the entire system.
