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It's basically the same idea in both cases. Power laws warp anything "slightly bigger" into dominating everything else when the power is big enough. There's a bit more stuff near the outside than the inside, so with a high enough dimension the volume is in the rind. Similarly, the equator is a bit bigger than the other slices, so with enough dimensions its surface area dominates.

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Yes, this seems to be the result of the standard Euclidean metric rather than the high dimension itself. I guess most people assuming the metric to be Euclidean, so it's ok.

The conditions you need for that to be true are substantially weaker than being Euclidean (though, when people are talking about "weird" behavior in high-dimensional spaces nowadays, it's in the context of ML and Euclidean stuff anyway). If you have a meaningful notion of dimension (basic properties like <1,0> being different from <0,1> and <1,0> being closer to <1,1> than <0,1>) and and don't have discretized shenanigans (which would collapse the inequality I'm exploiting to a strict equality, with some sort of 1^n=1 behavior) then the natural measure induced by the metric in question will exhibit the described behavior. You can easily verify that for every metric represented in scipy or whatnot, and a proper proof isn't too much more work.

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