E.g. would the solubility and Ksp of $\ce{Mg(OH)2}$ be the same in water as it would be in HCl? Doesn't solubility depend on the polarity of the solute and solvent, so wouldn't different amounts of a solute dissolve in different solvents. Wouldn't this give different Ksp values? However, I only ever see a single Ksp for anything.
Also, if $\ce{Mg(OH)2}$ is said to completely dissolve in $\ce{HCl}$, is it implied that is completely dissociates into $\ce{Mg^{2+}}$ and $\ce{OH^-}$ ions, or does it only exist as small solid pieces of $\ce{Mg(OH)2}$ floating around in the $\ce{HCl}$? I am assuming the latter, as the Ksp of $\ce{Mg(OH)2}$ is small.
I have a question in my book where $\ce{Mg(OH)2}$ is dissolved in $\ce{HCl}$ and then back-titrated with $\ce{NaOH}$. I can't seem to understand why, if the $\ce{Mg(OH)2}$ dissolves completely, we would need to do a back titration. I thought a back titration was useful when the solute doesn't dissociate completely.