As far as the QR core transportation is concerned, vulnerabilities cannot be stuffed into a QR code itself because it just encodes a query string (something like "address=bc1qabcdefg&amount=0.01&label=My address"), there is nothing an attacker can put in there to implant malware in Electrum, although they can of course change the address which'll generate a different QR code causing you to get scammed.
You did verify the checksum of the live ISO you downloaded, right? There have been cases of Linux distribution websites getting hacked and having malicious ISOs places instead. Somebody could place a package in the operating system specifically for snooping on Electrum private keys.
Sidechannel attacks are troublesome to do and as long as you close the curtains and don't let anyone gain physical access to your cold storage, I think that's sufficient and the difficulty of executing a sidechannel attack is still fairly high.
If you do an analysis on the power consumption of the computer, you could see a tiny spike in the power consumption of the device. That could leak the keys to the attacker. EM wave radiation, cold boot attacks all could pose a problem. I don't think any computer is specifically designed against that and even so, it would be difficult/impossible to remove that as an attack vector. I would be much more concerned about who would have access to the device than a side channel attack. I cited that as an example of how devices that are not specifically designed for such usage could have lesser safeguards (duh).
Different types of side channel attacks can be mitigated in different ways. For the attack that listens on keyboard taps and non-audible sound waves (
acoustic side channel attacks) can be defeated by playing audible white noise near the airgapped computer.
Cache side channel attacks that rely on reading the processor's internal cache don't work if you're not running in a VM, or some cloud server running in a VM, because such attacks need the memory pages containing Electrum to be mapped in VMs as shared pages. (Running the OS unvirtualized also partially breaks Meltdown from 2018, because it uses a cache attack after a race condition.)
In a timing attack you are trying to guess state by correlating it with execution time of that step, on the assumption that different speeds indicate different states. Slowing down the fast parts of ECDSA which derive private keys and make signatures for transactions to make all steps take the same time can prevent this but you'd need to update the libsecp256k1 library bundled with the OS. The performance drop shouldn't matter because Electrum is the only thing being ran on an airgapped computer.
Differential fault analysis that relies on heating up, overclocking or otherwise making the CPU unstable some other way can be prevented by not overclocking the CPU, and if it already has heating problems, replace it with a new one.
There aren't any known mitigations against power analysis attacks because Intel, and possibly AMD, provide opcodes for monitoring the processor's power consumption, and there isn't a way to disable that. This method requires an oscilloscope anyway so I guess if you see a big ol' oscilloscope sitting next to your airgapped machine, that should obviously ring some alarms. They only work from close range anyway, and I don't think they can even get close enough to the CPU because the desktop tower case is too big (hopefully the airgapped computer is a desktop).
So In short, you gotta:
- Play reasonably loud white noise
- Slow down libsecp256k1
- Buy a really big desktop case, and disable the kernel modules for USB and serial ports and everything else you don't need
- run Electrum on a physical OS
- Leave your CPU clock speeds alone
And that should protect you from most side channel attacks. Some of these are probably outside of your abilities though, it would be easier if someone made a live OS specifically designed for running Electrum.