Does oxygenating tap water remove the tap water flavor? I know flavor is subjective and not exactly quantifiable, but I think there's merit to this question.

We've all experienced that tap water often tastes a lot worse than bottled water or filtered water. I've always felt, however, that cold water actually tastes much purer than warmer water.

For instance, I was recently in a house where I knew the water was so called "hard water" and I could taste that it had the typical mineral flavor straight from the tap.

So, I took a glass of this tap water and refrigerated it for about a half hour and then drank it and I couldn't taste the typical tap water flavor at all.

I'd encourage people to try this just to make sure I'm not crazy.

Additionally, I have family that lives in Ohio and if anyone has ever tasted Ohio's tap water after living in a place like Washington (where I live) which has extremely fresh water, you know that Ohio's water is disgusting. I have found, however, that it tastes much better after being refrigerated for a while.

So, there are a couple of questions here.

Is it even true that refrigerating water (and thus cooling it) would cause a significant amount of oxygen to be dissolved in the water?

If so, would the amount of time the water is being refrigerated (even at a constant temperature) continue to dissolve more oxygen into the water? In other words, how long might it take for oxygen to reach it's equilibrium level of dissolution?

Those last two are definitely answerable chemistry questions. This next one is a little hairier.

Is there any reason, related to chemistry, that oxygen rich water would taste better than water not saturated with oxygen? Meaning, does the oxygen actually have a well-defined effect on the flavor of water, or is this simply answered by the fact that most people like the feeling of cold water more than luke-warm water and hence I identify cold water as "tasting better"?
Many public water systems spray water into the air or trickle over rocks (aeration) to let some dissolved gases bubble out (e.g. odorous $\ce{H2S}$). This also may make the water less acid and thereby precipitate dissolved manganese and iron.

When tap water treated with chlorine, bromine, ozone or other disinfectant is allowed to stand, open, then those chemicals also can escape.

However, I doubt anyone actually tastes dissolved oxygen (the human body can sense $\ce{CO2}$, but not oxygen, which is why miners kept canaries. In addition, one can taste the acid of $\ce{CO2}$ in soda water, and feel the bubbles bursting.)

So the practice of letting water stand and refrigerating is useful in improving flavor, but $\ce{O2}$ has little to do with it. Chilling water does increase the solubility of gases, though.