Regarding question 1, yes you can always ensure the image is closed.  You prove the strong Whitney by perturbing a generic map $M \to \mathbb R^{2m}$ to an immersion, and then doing a local double-point creation/destruction technique called the Whitney trick.  So instead of using any smooth map $M \to \mathbb R^{2m}$, start with a proper map -- one where the pre-image of compact sets is compact.   You can then inductively perturb the map on an exhausting collection of compact submanifolds of $M$, making the map into an immersion that is also proper.  

Regarding question 2, generally speaking if a manifold is not compact the embedding problem is *easier*, not harder.  Think of how your manifold is built via handle attachments.  You can construct the embedding in $\mathbb R^4$ quite directly.  Think of $\mathbb R^4$ with its standard height function $x \longmapsto |x|^2$, and assume the Morse function on $M$ is proper and takes values in $\{ x \in \mathbb R : x > 0 \}$. Then I claim you can embed $M$ in $\mathbb R^4$ so that the Morse function is the restriction of the standard Morse function.  The idea is every $0$-handle corresponds to creating an split unknot component in the level-sets, etc.  

Regarding your last question, the Whitney embedding theorem isn't written up in many places since all the key ideas appear in the proof of the h-cobordism theorem.  So Milnor's notes are an archetypal source.  But Adachi's *Embeddings and Immersions* in the Translations of the AMS series is one of the few places where it occurs in its original context.  You can find the book on Ranicki's webpage.