Like dissolves like generally refers to the fact that nonpolar compounds are usually soluble in nonpolar solvents and vice versa (polar/polar). I also take it to mean similar compounds dissolve similar compounds e.g. hexanes dissolve linear alkanes, H2O dissolves small, polar aldehydes and ketones, etc.
Benzophenone is somewhat polar, but it cannot H-bond. It does have a resonance structure with a negative charge on the oxygen which would allow it to interact better with polar solvents, but this structure is insignificant because it loses the stability from conjugation with the aromatic rings. Also, in general, H2O cannot dissolve large molecules very well, even if they are fairly polar. I can't say exactly why without a refresher. I IMAGINE (don't take my word for it), methanol could protonate the oxygen so make the bond more polar, so it would be more soluble in alcohols than in H2O. It is insoluble in hexanes because a) it is slightly polar and b) aromatic rings cannot form closeknit structures with hexanes, for example the way that hexanes and other linear alkanes could.
Biphenyl is insoluble in H2O for similar reasons; nonpolar, can't H-bond, large molecule. It is also VERY difficult to polarize because the aromatic rings are so stable. Methanol - not too sure how to explain that one. Hexane - biphenyl is "straighter"/flatter than benzophenone so it can probably more easily intertwine with hexane, if that makes any sense. Was it soluble or slightly soluble. That is an important difference.
Malonic acid: H2O - malonic acid can H-bond. This is very key for H2O solubility, pretty much everything that can H-bond is soluble in H2O. Malonic acid actually has two sites for hydrogen bonding. Methanol - same thing, can H-bond. Hexane - malonic acid is very polar, hexane is not. It also doesn't have a very long alkyl chain in between the carboxylic acids.
Feel free to correct me if you are more knowledgeable! But I think most of what I said is correct.