Might the universe be really small yet seem infinite from spherical reflections? if a single source of light quite small, was placed somewhere inside a completely reflective sphere, quite large, what would the reflections look like from the light source's point of view within that sphere?

:)

extra-credit: what would it look like over the course of time with movement starting from the moment the light turned on?

Found this question has been asked over in MathOverflow, sum clue?
The simple answer to your headline question is: No.

That kind of correlation between different angles has been investigated, and no significant correlation beyond 1 degree (first acoustic peak), e.g. large-scale circular structures in the CMB, has been found.

Details about the investigation of non-trivial topologies within the limits of recombination, see this Planck paper.

Closer-by structures (like galaxy clusters) visible by telescopic observations are correlated to gravitational lensing of the CMB, as described in this Planck paper, or by the integrated Sachs-Wolfe effect, as described in this Planck paper, since galaxy clusters tend to be correlated with dark matter halos.

Repeating pattens (from mirroring) would be evident from 3d maps (like this one for the local 300 Mpc environment) of the mass distribution or telescopic observations.

Multiple images of distant objects are well-known, e.g. as Einstein crosses, or Einstein rings, caused by gravitational lensing.
This can include mirroring in extreme cases; it's well-known how this would look like.