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Is there a single solvent that can dissolve chemicals/elements that when burned, will produce the color of its respective chemical/element?
If so, what chemical/element can be dissolved in this solution and burned to produce vibrant colors that can correspond to any of the following colors:
A hue formed by Blue and Green OR Pink and Violet will also work.
A single solvent? Probably not. Different elements have dramatically differing solubilities in most solvents, and so you would have to tune the solvent to the element at least to some extent.
However, for the specific task of identifying a solvent that would (a) burn and (b) dissolve chemicals suitable for giving either a blue/green color or a pink/violet color, then yes, I think it should be feasible.
Aqueous ethanol solutions at a concentration above about $20\%$ w/w are readily flammable at room temperature, and the aqueous base should provide a suitable base solvent for many salts. Vodka might be one good readily-available source of an ethanol-water solvent for this purpose.
To achieve the pink/purple color, potassium chloride should be a good option, as it's freely soluble in water and the potassium gives a purple color in the ethanolic flame.
Getting the blue color might be a bit trickier, as the most common element giving a blue flame is copper, and most copper salts aren't especially soluble in water at neutral pH, or in ethanol. What might work is to acidify the ethanol solution with a high-concentration vinegar stock (I've seen products available at up to $45\%$ acetic acid), and then add the copper salt. Copper sulfate pentahydrate is probably a good option there, as it's commercially available as, e.g., an algaecide. You'd have to experiment with the recipe to find something that burns readily with the desired color, but that doesn't smell too strongly of vinegar.
@ Joseph. If I were you I would start in advance by producing a regular vertical flame (nearly colorless) with a methane or propane or butane burner. A big stable flame, as long as possible. And then, just when you want the rest of the world know the signal, I would spray an aqueous solution of lithium chloride or of copper chloride horizontally through the flame (at the base of the flame). And repeat the spraying operation as often as desired. Each time it will produce a puff of colored light, which is as big as the length of the flame. It does not last more than half a second. But if you are two or three people equipped with such a spray near the flame, each one may send a puff one after the other. The emitted light will nearly look like a continuous red or blue-green flame. This is not dangerous, as only aqueous solutions are used.
