In this recent question, I learned that any two separable Banach spaces are homeomorphic. Based on some readings, I'm guessing that $L^2(\mathbb R)$ is homeomorphic to $\prod_{n=1}^{\infty} (0,1)$ (infinite product of the open interval, with the product topology).

Warm-up question: Is it true that $L^2(\mathbb R)$ is homeomorphic to $\prod_{n=1}^{\infty} (0,1)$?

(The space $\prod_{n=1}^{\infty} (0,1)$ is known as the pseudo-interior of the Hilbert cube $I^\infty:=\prod_{n=1}^{\infty} [0,1]$.)

Spaces locally homeomorphic to $I^\infty$ are known as Hilbert cube manifolds; spaces locally homeomorphic to $L^2(\mathbb R)$ are known as $\ell_2$-manifolds.

I seem to recall that the classification of compact Hilbert cube manifolds is very interesting: compact Hilbert cube manifolds are in bijective correspondence with compact CW-complexes, up to simple homotopy equivalence. But I'm not completely sure of the above statement, so I'll repeat it in my question below:

Questions:
• What is the classification of compact Hilbert cube manifolds?
• What is the classification of non-compact Hilbert cube manifolds?
• What is the classification of $\ell_2$-manifolds?

PS: Let's assume that all spaces are separable (or whatever implies Polish), to avoid pathologies.