Frequent BSODs (tcpip.sys and sometimes ntoskrnl.exe)

I am getting repeated BSODs, blaming TCPIP.sys and occasionally ntoskrnl.exe. The first recent one I can remember happened a few weeks ago when I installed ESET as part of a company-wide antivirus standardisation. Then, perhaps 10 days ago now, it started more regularly, a few times per day, to the point at which I can't do anything meaningful on my laptop.

I also noticed that any installer I downloaded directly onto that laptop would come up saying that it was corrupted, and wouldn't allow me to install. I've had success with one or two since then, but also started downloading to another device and then copying the installer across to the bad laptop to run/install.

I've been trawling forums and trying many different things, but so far, the BSODs continue.

What I've done so far, though I can't remember the order:

• did a system restore to the oldest point I had, BSODs continued
• did windows update after the rollback (system restore), 1 of the Windows Security Updates (KB2984976) fails repeatedly. Not sure if it's related, but did manage to install this manually.
• tried uninstalling the Realtek PCIe Fe Family Controller which an IT guy had said is incorrect for my machine. It blue screened a few times on the uninstall, or reinstalls itself on restart, but I think I just managed to get this unistalled and reinstalled now. Reinstalled from the Dell site: Realtek RTL8111E-VB Gigabit and RTL8105E-VB 10/100 Ethernet Controller Driver, Version 7.045.0516.2011,A01. An update for this is now showing under the Optional Windows Updates (though when I try update through Device Manager, it says the driver is up to date).
• scanned with the new ESET I had installed. Blue screened on more than one occasion just before finishing
• reinstalled Avast! which I'd been using before the change to ESET (did a proper uninstall of ESET first). Did a boot time scan, which came up empty.
• scanned with Malwarebytes - came up empty
• ran CHKDSK, it fixed up some things, but BSODs continued
• ran TDSSKiller, results attached

Right now, the BSODs happen frequently, and at varied times, like on startup, on running the scans, on trying to install windows on a VM, on trying to install/reinstall something etc.

Attached are the results of sfc /scannow I also just ran.