After spending most of the afternoon/evening trying to diagnose and fix this problem (wiping out everything, doing a complete system restore, etc.) to no avail, I've finally fixed the damn problem.
Going into the desktop version of IE, all I had to do was go to Compatibility View settings and UNCHECK "Use Microsoft compatibility lists".
Voila! Everything is back to normal.
All the sites I listed above loads normally again, I no longer get the "Your browser is out of date" notice on certain websites, it no longer takes 2 clicks on the back button to go back to Google search results, Instagram links are loading fine again.
I swear to god, this drove me baty INSANE. Sucks that I had to delete everything I had to get to the solution but what can you do.
Can anybody explain what just happened here and how I can potentially prevent similar browser mishaps in the future?