Remove Tarnishing/Hazing From Proof Coins

Actually in many cases you can. NCS does it successfully on many occasions.
Whilst their methods are proprietary, here is a little hint that you try on
Silver or gold coins only !
Lay the coin in a glass container add two ounces SUDSY AMMONIA and a few drops of water .
Leave in for two minutes . DO NOT RUB COIN !
Wash off with warm to hot water .
Repeat if necessary , let coin dry on a clean cloth or paper towel.

I would not recommend it . If you feel your coin is that bad just label it up as an impaired proof .

Does using distilled water avoid those concerns (about the presence of chlorine, fluoride, etc)?

Though, for info, even distilled water can be slightly acidic, by virtue of being in contact with air (even in the absence of pollutants).

My local coin shop uses E-Z-Est to dip them in. But it MUST be done very quickly with a rinse in distilled water afterwards. I learned this when I bought a proof silver Eisenhower dollar from him. His Ikes of the date I wanted all had haze. He took one out of the 2X2, dipped it quickly, rinsed it off, and showed me you could not tell anything had been done to it by comparing with a non-hazed proof of another year.

This dealer has a good reputation and years of dealing experience, owns Antietam Coin Exchange in Hagerstown, MD, has had "professional" grading training and employment by NGC in his past.

So he is not some newbie to all of this.

Hope this helps. But if you try, please practice on something else so you find that its better to do quick dips and rinse as even up to 5 seconds submersed can be bad.