1) If you want to get that technical, like fuel, battery cells aren't structural either. The battery casings like the fuel tank shells, are the structural components that can be used as structural elements. And on some planes, they are used in that manner for weight-saving.

2) Irrelevant. They're still more efficient than EV planes at any height, so comparing the relative efficiency of ICE to itself is pointless.

3) Neither can EV planes, unless someone has discovered how to turn gravity into electricity without water and a massive turbine.

4) ICE vehicles have higher power density (see previous Verge link, also https://www.topspeed.com/cars/warp-coils-seem-closer-to-real...). It's not even close. 43x-100x the power density of batteries, depending on the type of fuel and the type of battery. ICE vehicles are bigger because they are used to carry more things and people: hundreds of people or hundreds of tons of cargo on trips that can go almost halfway across the world without stopping. In contrast, an EV plane can currently carry itself, a pilot and a passenger, for almost long enough to get from LA to Fresno. (No EV currently on the market can even make the trip from LA to SF.)

And they don't have to to beat ICE in the market.

You're right. EVs will never be able to compete at the ranged market; they'll be limited to short-range hops. This means they won't be a viable option for the corporate/personal jet market beyond limited hobbyist use.