Generally speaking HA is a stable compound, however, pH can change its surface area and consequently its physicochemcial properties. acidic pH e.g. 2 and below or alkaline condition e.g. above 12 might change the HA properties. in addition, it is important to know how the HA have prepared, since its crystalinity, crystal phases and purity all affect on its stability in different solutions. I am not quite sure, but you may be able to check the conductivity of your solution as an indicator of the HA dissolution.
An interesting question! The choice of solvent is not immediately one I would consider immersing HAp in. I guess it would depend on how well the chloroform "wetted" the particles and what you intended to do with the system. Given the HAp would be completely insoluble in this solvent I guess you may not get anything happening like dissolution and ion exchange processes for which HAp is famous for would probably not readily occur in such a medium unless ions released could be supported in this medium.
Well dispersion of inorganic particles is always an issue and it affects the ultimate properties. You can go through attached paper where we used formic acid for dispersion (provided that your system allows this). Its the best solvent for dispersion of HA at nano level and provides stable dispersion even for days.