There are such spaces. A twisted sum a separable Hilbert space with itself will have cotype 2, logtype 2(hence type p for all 0

Yes, there is such a space as far as I can see.

The "Minkowski space" ℓ^p is actually such a separable Banach space. A separable Banach space is a Banach space in which there exists a countable, dense subset that approximates the entire space.

In the case of the Minkowski space ℓ^p the set of all sequences with only finitely many non-zero entries is a countable, dense subset. This subset approximates the entire ℓ^p space because every sequence in ℓ^p can be approximated arbitrarily closely by a sequence with only finitely many non-zero entries.

This is why the Minkowski space ℓ^p is a good example of a separable Banach space that has both Rademacher type p and cotype 2 for all 0 < p < 2, but is not a Hilbert space.

Please note: The term "Minkowski space" is typically used to refer to a four-dimensional space with a particular metric in the context of relativity theory, not for the ℓ^p spaces in functional analysis.

Yes

I'm looking for a single Banach space X that is type p for all 0