If I had to guess, I’d say that e1000 cards are pretty well supported on every public distribution/kernel they offer without any extra modules, but I don’t have any around to verify it. At least on this ubuntu I don’t find any e1000 related firmware package or anything else, so I’d guess it’s supported out of the box.
For the ifconfig, if you omit ‘-a’ it doesn’t show interfaces that are down, so maybe that’s the obvious you’re missing? It should show up on NetworkManager (or any other graphical tool, as well as nmcli and other cli alternatives), but as you’re going trough the manual route I assume you’re not running any. Mii-tool should pick it up too on command line.
And if it’s not that simple, there seems to be at least something around the internet if you search for ‘NVM cheksum is not valid’ and ‘e1000e’, spesifically related to dell, but I didn’t check that path too deep.