I'm trying to answer the second question. Most of the results can be found in the joint paper with Guilbault https://arxiv.org/abs/1712.05995.
Here I just wanted to make a quick summary.
An $m$-manifold $M^m$ with (possibly empty) boundary is *completable* if there exists a compact manifold $\widehat{M^m}$ and a compactum $C \subseteq \partial \widehat{M^m}$ such that $\widehat{M^m}-C$ is homeomorphic to $M^m$. In this case $\widehat{M^m}$ is called a (manifold) *completion* of $M^m$. One can change the homeomorphism to diffeomorphism or PL homeo. for other categories.

Dim = 0,1 are obvious.

Dim = 2, we can't find a complete classification in the literature, so we provided a theorem in that paper. That is, a connected 2-manifold is completable iff it has finitely generated first homology. This is mainly based on classical work of Kerekjarto and Richards.

Dim = 3, it's mainly due to Tucker, where he showed that a
3-manifold can be completed if and only if each component of each clean neighborhood
of infinity has finitely generated fundamental group.

Let me talk about dimensions $\geq 6$ first. Then I'll go back to dimensions 4 and 5. The first breakthrough regarding this problem was due to Siebenmann in 1965. In his PhD thesis, he proved that an open n-manifold $M^n$ is completable (it was called collarable by that time) iff
(1) M is inward tame, (2) the end is pro-$\pi_1$ stable and (3) the Wall finiteness obstruction of the end vanishes.
In 1983, O'Brien generalized the theorem to one-ended manifold with possibly non-empty boundary.

In our paper, we dropped the O'Brien's assumption on that manifolds are one-ended. By properly generalizing Siebenmann's conditions, we proved that manifolds of dimension at least 6 are completable iff they are inward tame, peripherally $\pi_1$-stable at infinity, of zero Wall and Whitehead torsion. Our proof is based on PL manifolds, but one can employ standard techniques such as "rounding off corners" to handle the other catergories.

Our theorem is still true in dimension = 5 provided that the fundamental groups are good in the sense of Freedman and Quinn.

The theorem fails in dimension = 4. Kwasik-Schultz and (independently) Weinberger discovered that there are open 4-manifold satisfying Siebenmann's condition but fail to be collarable.

Just a quick comment on Ian's reference to a contractible open manifold $M'$ constructed by Kister-McMillan which doesn't embed in $S^3$. The example was first proposed by R. H. Bing. Haken further proved that $M'$ doesn't embed in any 3-manifold using his finiteness theorem. Recently, I showed that $M'$ can't be embedded in any compact, locally connected and locally 1-connected metric space. https://arxiv.org/abs/1809.02628

openembedding into a compact one. I presume it's easy to embed any manifold as a locally closed submanifold of a sphere: just Whitney-embed as a closed submanifold of $\mathbb{R}^N$, send $\mathbb{R}^N$ diffeomerphically onto a ball, which sits inside a larger ball, and one-point-compactify the larger ball. $\endgroup$