2 typo fixed

I think the answer to your final question is yesno, and more generally: countable unions of embedded submanifolds are precisely the images of (not-necessarily-injective) immersions.

Sketch proof: A countable union of manifolds is a manifold, so a countable union of embeddings is an immersion. Conversely, by the Inverse Function Theorem, an immersion $f: M\to N$ is locally-in-$M$ an embedding; we thus obtain a "cover of the immersion by embeddings", and since manifolds are Lindelöf there's a countable subcover.

To answer your second-last question, we then need to analyse whether there are images-of-immersions that aren't immersed submanifolds (= images-of-injective-immersions).

(edit: fixed typo)

1

I think the answer to your final question is yes, and more generally: countable unions of embedded submanifolds are precisely the images of (not-necessarily-injective) immersions.

Sketch proof: A countable union of manifolds is a manifold, so a countable union of embeddings is an immersion. Conversely, by the Inverse Function Theorem, an immersion $f: M\to N$ is locally-in-$M$ an embedding; we thus obtain a "cover of the immersion by embeddings", and since manifolds are Lindelöf there's a countable subcover.

To answer your second-last question, we then need to analyse whether there are images-of-immersions that aren't immersed submanifolds (= images-of-injective-immersions).