If the Universe is Flat, has Finite Mass/Energy, and is Simply Connected, Then there MUST be an Edge, Mustn't there?
Yes, if the universe is:
flat (zero spatial curvature)
has finite mass energy (since we know it is uniform this also means it is bounded. If you drop the bounded es because you don't want to admit uniformity or otherwise, i.e., if it is unbounded, then the answer is clearly no)
is simply connected (has what is called a trivial topology)
Then it does have to have an edge.
See the zero curvature and other sections of the wiki article on the shape of the universe, it's fairly complete, at https://en.wikipedia.org/wiki/Shape_of_the_universe
The simply connected condition is critical also. If you allow other topologies then both the torus and the Klein bottle topologies are bounded, flat and have no edges.
There are a total of 17 possible different topologies for multiply connected spaces that are flat, in 3D (our spatial dimensions, which is what is referred to when one talks about curvature of the universe) Riemannian space. See fig. 4 in the arXiv paper at https://arxiv.org/abs/0802.2236 for all of them. There are others if the space is not flat.
As far as space being unbounded but mass energy finite, that would violate what we know of the homogeneity and isotropy of the universe. From the CMB we see the (large) scale homogeneity and isotropy. Now, we only see back to 380,000 years after the Big Bang, but no sign of large inhomogeneities. It could theoretically still be true that out inflation bubble is homogeneous, and thus the part of the universe beyond our particle horizon might not be, but there is no theoretical reason to think so. The more prevalent view is that it was as uniform more or less, and the same inflation that created our bubble might have created others. If we ever fully understand our inflation (which at this point looks pretty consistent with observations but those don't rule out various versions, or other unknown mechanisms from an unknown theory of quantum gravity), we might find out better or differently. But presently, a large scale homogeneity with possible bubbles is consistent with all observations.
First, it is not known that the Universe is finite. The WMAP experiment makes it more probable, that it is infinite. It was a satellite to measure the curvature of the Universe by finding tiny disturbances in the Cosmical Microwave Background. The result was zero (in the measurement precision).
Second, it is not known that the Universe is simply connected. For example, if the Universe has a constant positive curvature (which is smaller as the precision of the WMAP experiment), the Universe can have a spherical geometry. In essence it would mean that we live in the surface of a 4-dimensional sphere. In this case, the Universe doesn't have an edge, although the 4-sphere is simply connected and finite.
Third, it is not known if the Universe is finite. We can see only structures with our telescopes, whose light started at the Big Bang (they seem 13.7billion light year far, although they are currently around 30-40billion ly away because of the expansion). On such very high scales, there are no significant visible disturbances, thus we can think, the Universe out of this horizon looks probably the same (although it is not an experimental validation).
There is no data, what is out of this horizon, but the Universe is infinite and looks similar as in our direct environment, then there is infinite mass in it.
Furthermore, even if it has a planar geometry, it can have some cyclic topology. Imagine old computer games, as the player leaved the screen on the right and then appeared on the left. In this case it again doesn't have an edge (and it is not simply connected).
We can see the Universe not only in space, but also in time. The spacetime of the Universe has a singularity in the Big Bang. That we can see as an edge - but not in a far point, but in the far past.