Those large negative error codes with Internet Recovery Mode involve some sort of network related issue....unfortunately that is all Apple tells us. The issue can be due to your WiFi, local home network, your ISP, or your local Internet. Connecting the computer directly to the router at least provides a faster more reliable network connection which bypasses any difficult to troubleshoot WiFi issues.
As for the DFU Firmware Revive & Restore option......you need to be using a USB-only cable connected to the correct port of the Mac that is in DFU Mode. For an Intel MBAir, this DFU Mode port is the front left USB-C port. As for the USB-C cable, use the USB-C charging cable which shipped with your laptop from the factory.
Another common issue I encounter is with the USB-C ports themselves....many times they have accidental liquid damage (it only take one small drop that splatters & makes it into the USB-C port). And it is very difficult to actually tell if a USB-C port has any damage to it....even with my trained eye, I miss the damage at times.
Also contrary to Apple's documentation, I have found that the host Mac must be running the most recent major version of macOS which at this time is macOS 15.x Sequoia. If your host Mac (the one running Apple Configurator) is running any older version of macOS (or even an older version of Apple Configurator), then the DFU Firmware Revive/Restore will fail or not work at all.
If you still have issues, then you may need to take the laptop to an Apple Store or an Apple Authorized Service Provider for assistance and perhaps have them examine the laptop. I have very little faith in the 2018-2020 Intel Macs with USB-C ports & T2 security chip....their Logic Boards tend to have a much higher rate of failure than other older models. And the 2018-2020 Intel MacBook Airs definitely have some Logic Board issues...it doesn't help that they don't have a proper cooling system.....there is no heatpipe with the heatsink so the fan doesn't blow air over anything directly like in older models or in the MBPro models. I've just had to "retire" another one of my organization's 2020 Intel MBAirs for a bad Logic Board.
