Can ionic compounds can ever "dissolve" as molecular compounds do?
Can we refer to these free-floating ionic compounds as "dissolved" solute?


No. Dissolved implies that the ions are solvated. If the ions are in a solid ionic lattice then they are not solvated by the solvent and so would not be described as part of the dissolved solute.


  Does "XY(aq)" (the aqueous solution of an ionic compound) seem like a good description of this state?


Again no. $\ce{XY(aq)}$, in the context of ionic compounds, refers to $\ce{X^{n+}}$ and $\ce{Y^{n-}}$ ions solvated by water molecules. The dissociation is generally regarded as implicit. For example, $\ce{NaCl(aq)}$ implies an aqueous solution containing $\ce{Na+}$ and $\ce{Cl-}$ ions. A better description of the state you are thinking of would be $\ce{XY(s)}$, using state symbols, or even better you could describe it as a suspension.

From the IUPAC Gold Book:


  Suspension:
  A liquid in which solid particles are dispersed.


This accurately describes your system.