That's quite a missive you've written, we'll try to help you point by point, at least until you've begun to learn what's wrong.
I am trying to find out what I have created. I dissolved calcite (calcium carbonate) from some rocks I had in HCl.
Great. Do you know what reaction happened? If you don't, can you describe what you saw? It may you understand what's happening.
I was trying to recrystalize the calcite to a clearer and more aesthetically pleasing crystal structure from a super-saturated solution in an undisturbed location. I got impatient and boiled off the solution to get what seem to be yellowish (contaminated HCl) calcium chloride crystals...
Lots happening there. Briefly, mixing calcite with acid is not how recrystallizing is done.
Doing some reading, I found that neutralizing the solution with sodium carbonate may precipitate calcium carbonate again...
Do you have (or can you write) a balanced chemical equation for what you want to happen? It will he you know if its possible or not.
So I added some sodium bicarbonate instead (knowing that it can sometimes serve as a substitute in some reactions calling for Na2CO3). Nothing. I boiled it down and all I get is more yellow crystals with what seems to be a sodium carbonate coating.
Bummer.
Then I read something that said potassium could react with a calcium chloride solution... So I had some potash I extracted from ashes (which is least mostly potassium carbonate, if contaminated).
I wasn't really expecting a difference. I figured it would simply form a mixture if no reaction occurred. I had added a very small amount to it before and noticed that some pink crystals formed amidst the other colors when I boiled it dry. I assumed this was potash and noticed it was particularly resilient to dissolution in the HCl again.
Possibly. Again, a balanced chemical equation would help. You have to realize, the potassium won't disappear. So you have to expect that you're now getting a mixed batch of salts. And you've previously implied that mixing diverse reagents just "gets back" what you had before. You'll have to abandon talk like that, because that's not possible. For example:
Well... I figured I would try to make more of those crystals so I added nearly 10 ml of my improvised potash solution to the mixture of calcium carbonate, sodium carbonate, and HCl. Some minor bubbling, which I assumed was a simple neutralization reaction of the slightly alkaline potash with the HCl.
I don't know what you're expecting to get here.
The really strange thing happened when I tried to boil down this new mix. It nearly ran me out of my house. Boiling the other solutions produced some hydrogen chloride fumes, but it wasn't too bad since I am using a very small improvised beaker and I am only dealing with maybe a quarter cup of solution or less. I am somewhat used to muriatic acid fumes, but this was different. The initial boils resulted in an awful smell... Then much later, the smell was mostly chlorine-like. The solution is now forming brown crystals and still smells terrible when rehydrated and boiled.
I don't know what's happening here. Some free HCl may have completely evaporated. Just because you're used to fumes from HCl solution doesn't mean you're ready for all of the HCl to appear at once.
So my question. Did I make something toxic besides chlorine gas? I know hydrogen chloride and chlorine gas are both irritants in a high enough concentration. But this smell was different than either and much stronger. I wish I could boil whatever remains of it out of my mixture.
In summary...
Hydrogen Chloride (muriatic acid)
Calcium Carbonate
Sodium Bicarbonate
Potassium Carbonate
* Various trace contaminates
All of this has been mixed and boiled in various stages. Does anyone know any compounds which may possibly form from this? And why does it smell so bad when heated?
If you potash is made from soaked wood ashes, then you have some potassium carbonate and potassium hydroxide -- and a large amount of random organic compounds. Unless you've recrystallized the ash water repeatedly, and you won't get enough unless you've process a ton of trees.