1. I am not sure, but I suspect it has something to do with the atom's size. In organic chemistry we learn that nonpolar molecules like alkanes do not dissolve in water. That is, 'oil and water don't mix'. The reason is because water molecules have an exceptionally strong attraction for each other because of hydrogen bonding. This attraction is so strong they will link together into the form of a 'cage' around the grease molecules. Water molecules in this state have very low entropy because they are fixed in position. Higher entropy states tend to be favored over lower entropy states, so the fewest number of water molecules will act in this way as possible. The way to keep the number of water molecules in this low entropy state down is to keep all the alkanes in one blob; this way the surface area of the alkanes is minimized.
Something similar may be happening with iodine. It makes a nonpolar molecule. However, this explanation does not account for the fact that bromine and chlorine do dissolve in water, so there must be something wrong with it.
Actually, one might suspect iodine would dissolve better than the other two since it's more 'squishy' - that is, it's more polarizable.
Sorry I can't help you more with that question.
2. The only difference between K+ and Li+ is size for the most part. I suspect that the smaller ion will interact better with water because it's charge is concentrated in a smaller volume.
With Mg2+ and S2-, there are two differences that strike me immediately; one is the different sizes of the ions, and the other is their different charges. Water molecules have a single region of negative partial charge and two regions of smaller positive charge. How this affects the answer I don't know however. Sorry!
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