The answer to 2 is yes. Let $(X,d)$ be complete but not compact, then for some $r>0$ it has a countable r-separated subset $S=\{s_i\}:i=1,2,\dots$. Let $d'$ be the maximal metric on $X$ satisfying $d'\le d$ and $d'(s_i,s_j)\le r/\min(i,j)$ for all $i,j$. Then $(X,d')$ is homeomorphic to $(X,d)$ - in fact, the identity map is a local isometry - but $d'$ is not complete. So we have embedded the space into the completion of $(X,d')$ as a non-closed subset.
The metric $d'$ can be constructed explicitly: $d'(x,y)$ is the minimum of $d(x,y)$ and the infimum of sums $d(x,s_i)+d(x,s_j)+r/\min(i,j)$ over all pairs of $i,j$. Verifying the triangle inequality is straightforward.
As for 3, the answer is no, because you cannot embed any complete space into a compact space. For example, a non-separable Banach space cannot be so embedded, as Qiaochu Yuan explained in comments.
Update. It seems that I misunderstood Q3. As stated, it asks whether every compact space can be embedded into a complete non-compact one. The answer is of course yes, as Ady noticed.
Firstly, if you just have pairwise distances rather than coordinates of points in $\mathbb{R}^3$, then you can attempt to embed the mesh into $\mathbb{R}^3$ using Isomap. Under certain conditions, such as if you have a triangulated convex polyhedron, the isometric embedding into $\mathbb{R}^3$ is unique up to isometries of $\mathbb{R}^3$.
After you embed your meshes $X, Y$ as finite labelled subsets of $\mathbb{R}^3$, the problem of finding the optimal scaling and isometry which minimises the error $\sum_i \lVert X_i - Y_i \rVert_2^2$ is called Procrustes analysis. After applying this isometry, you get a canonical (root-mean-square) distance between the sets $X, Y$.
The way that Procrustes analysis usually proceeds is to subtract the mean, then scale to have unit variance, and then use singular value decomposition to find the optimal orthogonal matrix transforming $X$ to match $Y$ as closely as possible. The last part is called the Kabsch algorithm.
Best Answer
The answer was given in the papers of I. Schoenberg and von Neumann, MR1501980, MR1503439, MR0004644.