It was shown by the German mathematician Cantor that any mapping

$$ f:\mathbb N \rightarrow \mathbb R $$

cannot be surjecive, which means that there is no way to map a discrete infinite basis in a Hilbert space into a continuous one. This is rephrased as: a separable Hilbert space cannot be isomorphic to a nonseparable one.

It was shown by the German mathematician Cantor that any mapping

$$ f:\mathbb N \rightarrow \mathbb R $$

cannot be surjective, which means that there is no way to map a discrete infinite basis in a Hilbert space into a continuous one. This is rephrased as: a separable Hilbert space cannot be isomorphic to a nonseparable one. This means that there is a negative answer to all your questions.