[crf92]

I observed some ice that had formed from a puddle, and I became curious as to whether or not 32 degrees Fahrenheit is sufficient enough to make ice behave like a "true solid", per se.
So, I guess the question I'm trying to ask is: Is ice at the freezing point of water capable of having viscosity and other properties that are normally ascribed to liquids?

If I'm not being clear, please let me know. It's a very awkward question to phrase, haha.

[The Jar]

What you're asking about is actually about the nature of phase transitions.  It seems like you are wondering whether or not water transforms from something that is definitely ice to something that is definitely water continuously or noncontinuously.  You asked the question vaguely enough that there is no "yes or no" answer.  To the first approximation it is a noncontinuous transformation so solid water (ice)  is fully a solid even right before the phase transition.

However, there are actually some special properties of ice that gives it some surface "water-like" behavior above (I think) -40 degrees C or so.  However, by surface I mean the first few layers of atoms, so these are properties you won't directly observe on the macroscopic scale.  It is what makes skiis and ice skates work, though.

There is another, separate phenomenon called diffusion which is how quickly atoms move around, and this does happen in solids and actually appreciably fast close to the melting point of a material.  This is nothing like liquid fluidity, though.

This doesn't really come close to answering your question, though.  What you really want is to learn about is first order and second order phase transitions.  Also nucleation theory.  And sintering.  These are advanced topics but even someone with only general chemistry knowledge can understand the foundational ideas if they are presented properly.  Try an introductory physical chemistry textbook (maybe a sophomore or junior college level).  You'll understand how to phrase your future questions even better.