Ok that makes more sense. Although most Airspace apps don't use NSIS, Touchless For Windows does. You can ignore the Leap Motion installer.log, as it's clear that the Leap Motion Core Services already installed alright.
Let's first check if your TouchlessForWindows download to make sure it isn't corrupted. Could you make sure it's exactly 45,734,552 bytes? If you have an md5sum tool, the md5sum should be a210b409dd9e7ee42ff9a0cb58fbeee4. If it doesn't match, you can download a working build here.
As for Airspace Home not launching after system restore, that's likely a separate problem. You can first take a look if there's any useful info at the end of %LocalAppData%\Airspace\log.txt and paste that here. However, as this only happened after system restore, you might just be running on an earlier version of the software so it would best to upgrade and debug with Airspace version 1.2.
That "Error downloading EULA" text was removed in our most recent version of the software, so you probably also got this after system restore. I would recommend debugging this only with the latest version of the software at https://leapmotion.com/setup. Because version 1.2.0 upgrades the firmware, your embedded Leap Motion in the HP Envy won't run with version 1.1.3 and earlier versions of the software anyway.
Hopefully that gives some guidance on your three distinct problems.