Could you tell us exactly what these errors you get while playing say?
I'd think, if the installer got up to the point where it's trying to install level packs, then it's already downloaded and installed the game itself (it always does the game, styles and editor first; then levelpacks afterwards). So these errors likely aren't related to a broken installation - but we should still figure out what's going on.
Since you say you're able to manually download packs, I assume you know how to unzip ZIP files. If it really is a broken NL install, you could try doing a manual install instead of using the installer. You'll need to download these two ZIPs, and extract them to the same folder. The third download here is just a single file, which should be saved to the same folder the ZIPs are extracted to.
Make sure the folder you extract them to is not one that requires run-as-admin to write to, as NeoLemmix assumes it can freely create / modify files in its own folder.NeoLemmix V12.7.3:
https://www.neolemmix.com/download.php?id=372Styles and sound effects, November 2019*:
https://www.neolemmix.com/download.php?id=349(Optional) Level editor V1.17:
https://www.neolemmix.com/download.php?id=350* December 2019 styles update hasn't been released yet. Keep an eye on the Graphic Sets board for the release - Nessy usually posts a ZIP that can be used as an "update"; this ZIP again is just extracted to the same folder as everything else.For level packs, we encourage creators to ZIP them in such a way that they, too, can simply be extracted to the main folder. However, not every level creator does this, so look at the contents - if the level pack ZIP contains "levels", "styles" etc folders, it extracts to the main NL folder. If it just contains a folder with the pack's name, you'll need to extract it into NL's "levels" folder instead, etc.