What is the interaction between dissolved ions of opposite valence in a solution at rest?
The answer is: it depends. Dissolution of a salt implies that the entropy gained exceeds the cost of breaking lattice interactions (the solution enthalpy, assuming it is positive). Electrostatic interactions compete with kT (thermal jostling). Under physiological conditions, long range interactions are strongly screened by intervening solvent molecules and other ions. At best ions are subject to an effective potential due to distant ions, in particular between segregated charges as in the cases you present (e.g. on opposite sides of a membrane).

However, ionic solutions are generally regarded as "non-ideal". This means that interactions between the ions and with the solvent cannot be dismissed. At high salt concentration ion pairing will be encouraged. Above the solubility limit salt precipitation will occur, preceded by formation of clusters (seeds). If an ion has multiple charges, then electrostatic interactions will be stronger and pairing will be encouraged. Some metal ions form permanent ligand complexes that can attenuate the overall charge or distribute it over the complex, and the ligands can form a "cage" about the central ion, altering the interaction potential with solvent and other ions. The effect of temperature is complex. The entropic cost of association increases with T, but the dielectric constant of the solvent tends to decrease also.

If you are interested in reading on ion-pairing in NaCl solutions you may want to start with Ref. 1. It explains that association is negligible in dilute solutions:


  For dilute NaCl, we predict a coordination number of 5.3 with only waters in the solvation sphere for Na+ at 25 °C, 1 bar. With increasing NaCl molality, we find increased Cl complexation of Na. At 25 °C and 1 bar the Na+ ions in an 8 m solution of NaCl have 4.3 waters and 1.0 Cl ligands in their first coordination shell. There is some evidence of solvent-separated Na–Cl ion pairs from the peak in the Na–Cl pair distribution function at 5 Å. However, this disappears with increasing temperature. 


There is more on formation of NaCl complexes and clusters in that article.

References


Sherman, DM Collings, MD. Geochem. Trans., 2002, 3(11), 102–107