Did you first try working out what happens for one-variable discrete harmonic functions? You said that "such functions are clearly more constrained than their one-variable counterparts" but must have meant one-variable functions that are real-differentiable rather than harmonic. For one-variable discrete harmonic functions are very constrained:
If you know $f(x+1)$ and $f(x-1)$ you can first interpolate $f(x)$ and then extrapolate $f(x+2)$ and $f(x-2)$. By induction, every value of $f$ is determined this way. That settles the special case of the standard suite of theorems (minimum-maximum principle, Liouville's theorem, mean-value theorem, etc) for a unit interval.
You can piece together unit intervals to prove the more general case. That piecing-together logic is similar to the proof of Green's theorem that reticulates the domain and passes to the limit, so you could say this extends to the infinitary case(, modulo analytic due diligence)diligence.
As for smoothness (harmonic functions are actually real-analytic), I'm not sure of the proper discrete analogue, so I can't help you there.