What I kept reading is that chalk will work but can't be dissolved easily.
If by 'works' you mean it works as an alkali, yes that it true.
CaCO3 + 2H+ --> Ca++ + 2HCO3-
i.e. it absorbs protons (H+) to become the conjugate base, HCO3-. If you mean it reacts to release calcium ions - yes that is true too. The reason it is not effective as a means of adding calcium to water or in neutralizing acid in the mash is that the dissolution reaction is slow at normal mash acid concentration and pH. A dose is added (and most of the spreadsheets calculate a dose many times larger than is actually needed) and this is added to the mash but the reaction takes place so slowly that by the time the mash is over there is still lots of undissolved chalk left in the mash. The brewer reads a pH he likes and proceeds with the brew but that additional chalk keeps dissolving and reacting pulling mash pH too high and ruining the beer (it will be drinkable but not nearly as good as it could be if mash/wort pH is managed properly).
Seltzer water is usually just carbonated water but can be harder to find. I make my own carbonated RO water with a Sodastream Soda maker. Club soda will have about 200mg of sodium in a one liter bottle and a tiny additition of Bicarb to buffer it.
No bicarb is added. Some of the dissolved CO2 is converted to bicarbonate. For example, if you carbonate pure water to the extent of 2 volumes the pH would be 3.7 and the water would contain 12 mg/L bicarbonate. One could, of course, add more but this would defeat the purpose.
This should make very little difference but should be accounted for if sodium is a concern.
As I noted earlier the sodium chloride is there for taste. Without it the carbonated water tastes relatively flat and thin. The salt is in there for the same reason old timers put salt in their beer.
Add some baking soda to a glass of water, too salty but it will buffer the acid in your stomach. Most people add a teaspoon to a glass. I have been adding about a tsp to 5 gallons.
I often advise people to put some bicarbonate in a glass of water and taste it before adding it to their brewing water. It tastes pretty bad to most people but I do have to say I find it particularly offensive. If you like it, then go ahead and do it but as I have pointed out, I think this is the third time, it will leave the water when mash pH of about 5.4 is established.
I have read that as you lower the temperature of water, chalk actually becomes more soluble.
That is true. The solubility product (of calcite - the least soluble form) is 10^-8.45 at 20 °C but 10^-8.39 at 5 °C. This means that calcite is 10^0.03 = 1.07 (i.e. 7%) times more soluble at 5 than at 20.
The water can become saturated at that temp.
The water can be saturated at any temperature. The water is saturated whenever {Ca}*{CO3--} = solubility_product and super saturated if greater.
When you heat the water gently, the chalk can remain dissolved and is said to be super saturated at that higher temperature, but still dissolved. Now when boiling a super saturated liquid you will cause a precipitation.
Carbonaceous waters, especially from wells in limestone regions often leave the tap supersaturated with respect to chalk and/or carbonic acid or both. It often takes days for equilibrium to be reached unless something is done to hasten this. 'Something' usually consists of
1. Adding nucleation sites (some chalk)
2. Heating
3. Raising the pH
4. Adding additional calcium
5. Sparging with steam or air
6. Combinations of the above (such as adding lime which combines 3 and 4)
Would a couple of grams of chalk dissolved in a liter of seltzer water added to 5 gallons of water be considered super saturated?
Assuming 'a couple' to mean 2 when added to a liter of water and dissolved with CO2 to pH 7 then the liter is supersaturated with respect to CO2 (0.16 atmospheres with the atmosphere itself at 0.0003 atm partial pressure of CO2) and chalk (saturation pH is 5.7 - we set for 7). Added to 5 gallons of pure water the solution is supersaturated WRT CO2 (0.01 atm) but not WRT chalk (saturation pH 7.8 and we set pH 7). Now because the water is super saturated WRT CO2 CO2 will leave it until the partial pressure of CO2 equals the partial pressure of CO2 in the atmosphere (0.0003 atm). At this point the pH will have risen to 8.52 which is well above the saturation pH and calcium carbonate will precipitate. This would, at room temperature, take a long time. Adding heat will accelerate the process and a precipitate will form.
I'm not seeing a precipitate when heating, only clear hot brewing liquor.
Two grams of calcium carbonate in 5 gallons (105 mg/L) dissolved with CO2 to pH 7 yields water of, surprise, hardness 105 and alkalinity of 105. This is definitely super saturated but if you heat this water short of boiling, and don't provide nucleation sites it is possible for much of the carbonate to remain in solution. My well water is about at this level of hardness and somewhat less in alkalinity and doesn't appear milky or drop a precipitate unless I do the other things. However if I put a straight (no nucleation stuff) sample in a nephelometer pre-boil it reads 0.57 NFU and after a brief boil 1.97 NFU so the precipitate does indeed form. The particles are obviously tiny and will take forever to settle but settle they eventually will as equilibrium hardness and alkalinity with atmospheric CO2 are 50 and 50 each.
So the bicarb from the temporary hardness, the baking soda and the chalk dissolved with CO2, buffers the PH of the mash dissipating as CO2 and leaving sodium and calcium.
If it's dissipating it is not buffering. As I mentioned before the carbonic/bicarbonate system buffers at 6.38 (at 20 °C), not mash pH. Bicarbonate stresses the natural buffering of the base malt which without that stressor would be at 5.65 - 5.75 nominally depending on the malt. The stressor raises mash pH above what is desired.
That sounds bad to me unless there is acid from dark malt trying to stress the buffer in the other direction (low pH). In those cases you need some alkali and bicarbonate will do. But so will OH- from lime. It seems simpler to me to add some lime to the water/mash in those cases but if you prefer to use chalk and CO2 that's fine.
If you brewed two beers, identical in every way except one had a mash PH of 5.2 and the other a mash PH of 5.5, can you tell the difference?
I've never gone as low as 5.2 but I can certainly attest that the difference between beers brewed at pH 5.4 - 5.5 are dramatically better than ones brewed at 5.6 - 5.7, at least for the styles I do (lagers, German ales).