Application of unexpected theorem "all closed manifolds are a quotient of $\Bbb B^n$"! Recently just accidentally I read a corollary in Lee's Introduction to Riemannian manifolds which surprised me:

Corollary 10.35 (LEE- IRM). Every compact, connected, smooth $n$-manifold is homeomorphic
to a quotient space of $\overline{\Bbb B}^n$ by an equivalence relation that identifies only points on
the boundary.

To me this result is so much well-set and less in assumptions and restrictions. But has this result any application? I mean how this can help in differential geometry theorems as well as its face shows that it is so powerful?
My second questions is this: Can one produce all closed smooth $n$-manifolds by equivalence relations on $\partial \overline{\Bbb B}^n\simeq \Bbb S^{n-1}$ that come from some group actions? I mean can one represent or identify that equivalence relations by some (topological) group actions?
 A: On the second thought, yes, there is one application of this result: The classification of compact connected surfaces usually (not always) starts by proving that every such surface is the quotient space of a polygon (by an equivalence relation induced via homeomorphisms between pairs of edges). But in higher dimensions the classification (in any sense of the word) does not use this theorem and proceeds along very different lines.
In higher dimensions, the result in Lee's book holds even for topological manifolds, but the proof is much harder:
Brown, Morton, A mapping theorem for untriangulated manifolds, Fort, M. K. jun. (ed.), Topology of 3-manifolds and related topics. Proceedings of the University of Georgia Institute 1961. Englewood Cliffs, N.J.: Prentice-Hall, Inc. 92-94 (1962). ZBL1246.57052.
I checked MathSciNet: It lists 18 papers referring to Brown's paper. Feel free to check them one-by-one. Several deal with topological dynamics on manifolds. I never heard of any of these papers before and did not find the titles of any of these papers particularly interesting.
As for your last questions: Quotients by non-proper group actions tend to be non-Hausdorff. If you quotient by an action of a compact Lie group of positive dimension, dimension drops. This leaves us  with finite groups. Armstrong's theorem implies that if $G$ is a finite group of homeomorphisms of $S^n$, then $S^n/G$ is simply-connected. There are probably even more topological restrictions, but this already shows that you will not get all compact connected manifolds this way.
