While I'm not exactly the kind of chemist that would be able to tell you a full story on beer water chemistry, I'll try.
It appears to be pretty straightforward, but does require some basic chem knowledge. This tool allows you to see the water profile you're aiming for, and to enter the initial salt concentrations in your water. Then, it shows you how it would change by adding a certain amount of each salt. It looks like it also tells you how much of the salts you're adding into the initial mash and into the sparge water, depending on the volumes you're using. The main focus seems to be on residual alkalinity and Cl:SO4 ratio. It gives you the option of using Hydrochloric or Lactic acids to adjust it, but I think most people just use 5.2 buffer and don't worry about the residual alkalinity and pH because it'll keep it where it should be to cover just about almost every beer you can brew. I don't know anything else on the matter though. Cl:SO4 seems to affect the balance Cl = more bitterness + clarity and Sulfate = more malt. So by playing with the salts, you can get your desired balance. It also shows you what effects each salt has on the alkalinity, but again, I don't know more about it.
I don't know if this is at all helpful, because even I can hardly understand what I just wrote. I really do suck at explaining these kinds of things in forum settings. As for Cl vs NH2Cl, I think NYC still uses Cl. At first I was a bit paranoid, and used RO water to remove everything from it, but now I just go with straight tap water. If there is only Cl, it'll leave the solution and evaporate as you're heating your water, and your beer will be fine. If there is NH2Cl, it won't leave the water and your beer will have funky off-flavors. Since none of my beers exhibit them, those of you who had them would know, I figured we have just plain chlorinated water.
As for filters that remove Na, SO4, and Cl by attaching to the tap, I don't know. I do know that activated charcoal ones (like Brita or Purr or something like that, check what they say) do remove chlorine. When I doubt, use RO
That'll get you as close to distilled as possible without distillation or multi-thousand dollar lab grade filtration units. As for removing alkalinity, it is not something that you can just take out or add. Alkalinity is the ability to neutralize acids and depends on the salts that are dissolved in it. So depending on what salts you remove or add, your alkalinity will shift up or down.
Hope that helps a bit, though I'm not sure it will... now, a fresh pint would definitely help!