That's a hell of an "if" statement there my friend :)
I agree though, in a thousand years there will be a worldwide electricity grid that takes care of the "big picture" energy flows; hydrogen will be created as needed to serve as chemical feedstock to make eg plastics or hydrocarbons for specialized purposes. That or we kill ourselves in the meantime, but since killing everyone is in nobody's best interest I don't think that'll happen. Hopefully.
Indeed! It's one of many reasons that "the best" isn't always the real best.
I'm extremely optimistic about technology, but not so about politics.
Let's take one half-way around the world.
That's 1 sq m * 20000 km = 20000000 m^3. Let's take aluminum because it's cheap. 2.7 tons/cubic meter = 54 Mt.
Multiply by price: $2444 x 54M = $132B for just the wire. That's $16.5 per capita.
But you also need insulators, labor, machinery, design, and so on.
[0] https://en.wikipedia.org/wiki/List_of_countries_by_primary_a...
If you are willing to increase mining, and want the cheapest solution, and you can ignore or solve geopolitics, the “best” solution is neither hydrogen or batteries, it’s to make a several square meter cross-section HVDC loops around the planet, because nighttime winter is never more than 20 Mm from daytime summer and the losses are low enough that it is still worthwhile.
But: you probably still want hydrogen and batteries for the vehicles, and vehicles use so much power that solving transport almost automatically gives you enough used parts to build the storage capacity for the electricity grid, and that even with batteries (and over-provisioning PV, which is fine because of how cheap it is), there would be enough storage for winter.