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This must be why kinds (types of types) in Haskell are a separate and less powerful thing than ordinary types?

I believe it to be historically true, but Dependent Haskell might change this (https://ghc.serokell.io/dh see unification of types and kinds).

In Lean (and I believe Rocq as well), the Type of Int is Type 0, the type of Type 0 is Type 1, and so on (called universes).

They all come from this restriction.

With respect to Lean/Rocq, that's true, with the subtle difference that Rocq universes are cumulative and Lean's are not.

I suspect not, because in that case Type is not a Type itself, but a Kind.