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.