It a famous result of Maurey that a Banach space $E$ is finitely representable in $X$ if and only if it is a subspace of some ultrapower of $X$. Is there an analogous result for complemented subspaces? More specifically,

Let $E$ be a Banach space with an FDD $(E_n)$. Suppose that $E$ is isomorphic to a complemented subspace of some ultrapower $X^U$. Are the subspaces $(E_n)$ uniformly complemented in $X$?

It's a famous result of Maurey that a Banach space $E$ is finitely representable in $X$ if and only if it is a subspace of some ultrapower of $X$. Is there an analogous result for complemented subspaces? More specifically,

Let $E$ be a Banach space with an FDD $(E_n)$. Suppose that $E$ is isomorphic to a complemented subspace of some ultrapower $X^U$. Are the subspaces $(E_n)$ uniformly complemented in $X$?

Every space finitely representable in a Hilbert space is isomorphic to a Hilbert space. Also, every subspace of a Hilbert space is again a Hilbert space. Can we have a partial converse to this?

Suppose $X$ is a separable Banach space such that every space finitely representable in $X$ is a subspace of $X$. Must $X$ be isomorphic to a Hilbert space?

It a famous result of Maurey that a Banach space $E$ is finitely representable in $X$ if and only if it is a subspace of some ultrapower of $X$. Is there an analogous result for complemented subspaces? More specifically,

Let $E$ be a Banach space with an FDD $(E_n)$. Suppose that $E$ is isomorphic to a complemented subspace of some ultrapower $X^U$. Are the subspaces $(E_n)$ uniformly complemented in $X$?