Why is there an increasing solubility in water for chlorides, chlorates, perchlorates in that order? According to Wikipedia $\pu{100 mL}$ of water dissolve at $\pu{25 ^\circ{}C}$ about $\pu{35 g}$ of $\ce{NaCl}$ (1), but about $\pu{79 g}$ of $\ce{NaClO3}$ (2), and around $\pu{210 g}$ for $\ce{NaClO4}$ (3).

This looks to me roughly like its doubling solubility for every single higher level of oxidation.

Earlier I was under the impression that the % of $\ce{Cl}$ or % of $\ce{Na}$ in the water would still be the same, and the amount of additional oxygen is what could be causing the increase in solubility.... ie. now each molecule has 3 oxygen atoms as opposed to none in case of $\ce{NaCl}$. But this doesn't hold true.

I would also like to know what is the effect of multiple solutes sharing same atoms on each other.  For example does dissolving some chlorate cause the solvent to hold more chloride ?

As you may have guess Chemistry isn't my strong suite, I know basic chemistry, but now when I am looking at it again.. the whole thing has changed atleast w.r.t what I was taught during my school days. (right from how we determine the number of electrons)
Science is all about discovering hidden regularities, patterns, and laws. In that sense your bold generalization is good, but will it hold if we check just one more data point? Let's see: $\ce{KCl}$ has solubility of 25g/100ml, $\ce{KClO3}$ has 8, and $\ce{KClO4}$ has 1.5. Er, well...

What we have here is an interplay between solvatation energy and lattice energy, the latter being determined by the crystal structure, which is a tricky thing, and hence so is solubility. Don't expect it to be predictable from qualitative considerations. You don't know it until you measure it. Chemistry is an experimental science, after all.