Is it correct to think about a point in time as the set of positions of all "things"? Is it correct to think about a point in time as the set of positions of all "things" (photons, electrons, etc) that exist in the universe at that moment, despite the fact that simultaneity is relative?
 A: When we talk about positions of events in the universe this assumes we have chosen some coordinate system that we are using to locate spacetime points. This might be a Cartesian system $(t, x, y, z)$, polar coordinates $(t, r, \theta, \phi)$ or some more esoteric system such as the Kruskal-Szekeres coordinates.
What you describe is choosing a coordinate system then identifying the 3D submanifold $(t = \text{constant}, x, y,z)$ for some constant value of the time coordinate $t$.
This broadly matches your definition, though be a little cautious about saying:

set of positions of all "things" (photons, electrons, etc) that exist in the universe at that moment

A particle (classically at least) is located at a spacetime point, but there are an infinite (transfinite?) number of spacetime points that don't have anything located at them. Also remember that the constant time we've chosen is just the time in our coordinate system. Other observers making different choices of coordinates will not necessarily agree that the time is constant for our submanifold.
