Well, to be candid: Isn't that slightly moronic and totally irrational of your higher-ups to break your workflows on the extremely slim chance of a an artificially inflated (by Apple), yet infinitesimally likely potential security issue? Notice the number of conditional phrases... Seriously, this is stupid beyond belief and clearly shows that the people in power don't actually understand the (non-)issue at all. Otherwise I'm not sure your post makes much sense since you haven't provided any proper technical info like version of AE, version of Windows, system specs, output settings and so on. That aside, just removing QT should have no influence on other Windows components, since it's a completely independent infrastructure and to me it seems that this is just another indication of your "powers" not knowing what they are doing. They probably destroyed your audio CoDecs and audio driver, which as a minor will necessitate an excursion to the device manager, removing this stuff and letting Windows repair-install it.... Really, I'm sitting here not knowing whether to cry or laugh about the matter. It's so ridiculous.
You can install QuickTime components without the player or web plugin that had the (tiny) risky part in them. I'm with Mylenium, your powers-that-be are being silly. I've found many times that knowledge isn't power - in fact, it often eludes those in power...
If they won't let you have QuickTime (and they totally should let you have it), you may need to uninstall and reinstall your Adobe CC stuff. That should get stuff back working again (for the most part). Adobe did make a blog post about adding some native QuickTime decoding into future versions of AE, but that's not going to help you now...
To your other issue:
You can still make an H.264 without QuickTime by sending it to the Adobe Media Encoder. The QuickTime H.264 option was buggy anyway and Apple didn't even support it, so I would recommend against using it even if you had QuickTime.