Windows freezes shortly after startup

I also just checked the HDD with SeaTools (CHKDSK was fine), it found issues and told me to use SeaTools DOS to attempt to fix them, but that doesn't detect the HDD at all when I start it.

Use the bootable Seagate Seatools CD

The only way to get internet access is using VPN because I'm in my universities network. Only plugging in the LAN cable works fine but doesn't allow me to access anything beyond the internal network of the university.

As for SeaTools, that's exactly what I did, I booted it up but it didn't detect the drive.

So I've been without internet for a few days and the laptop has been running just fine, perfomance as always and no crashes whatsoever.
Right when I got into my homes WLAN network it attempted to connect and crashed again, so I can pretty safely assume that the internet connection causes the crashes. I've also tried direct connection via LAN cable as well as USB tethering with my phone and all of them cause the same crash.
All steps in the troubleshooting guide have failed up to the repair install so far, so I wanted to try the Repair Install but I get stuck at step 9 because accessing the internet again causes the same crash also in the setup screen. Would it be safe to skip the updates during the installation and install them afterwards after the issue has (hopefully) been resolved?

Yes, but I'd do a Clean Reinstall - Factory OEM Windows 7 to get shed of the Samsung bloatware and duplicate utilties.

Plug in by ethernet during install to see what driver Win7 provides in installer, or import the one on the Samsung Support Downloads webpage for your model. Then go online, enable Automatically deliver drivers via Windows Update (Step 3) and install all rounds of Important and Optional Updates with reboots. Keep checking for more Updates. When there are no more Updates, which drivers are still missing in Device Manager?