The substitution ratio for mini marshmallows to large marshmallows

Marshmallows expand so much because the water in them becomes steam, and gas takes up a LOT more volume than liquid. Specifically, 1 mL of water becomes ~1.36 LITERS of vapor, before it gets heated further. That's 1000-fold expansion, before you add additional expansion as the gas is heated.

Marshmallows don't have all that much water content, but when it's trapped in a stretchy gelatin matrix that holds gas readily, it only takes a bit to blow the whole thing up like a balloon.

You are correct that the gas expansion on its own is insufficient; unless I've badly muddled my calculations, gas expansion from 20C to ~150C (caramelization temperature) will increase the gas volume by under 50%.

It's exactly your deviation which caused the problem. Stiff meringue weeps - the foam slowly loses the egg white moisture. When you mix cornstarch in it, the cornstarch absorbs the water droplets before they have the chance to make a big wet spot. So if you use the marshmallow mix, you won't have this problem.

As for crystallization, I am not sure that xanthan gum helps. It prevents crystallization in ice cream, but there you have too much water between the dispersed particles of the other ingredients, and this water forms ice crystals. Xanthan thickens the water and makes the water droplets smaller, so they don't freeze to big, crunchy crystals. In a sugar syrup, you have it the other way round. Sugar is better soluble in hot than in cold water, so once the syrup cools down, there is too little water and too much sugar. I don't think that xanthan is capable of doing for the sugar what it does for the water. But even if it helps, it will make your result firmer than you want it, you'd have to reduce the gelatin, risking to end up with marshmallows either too firm or too soft.

The usual way to prevent sugar syrup from crystallization is to add an acid to it. This splits the sucrose molecules into glucose and fructose, which don't crystallize at the saturation sugar does. Be careful what you add, you don't want your marshmallows to smell of vinegar. Cream of tartar is the neutral smelling choice, and citric acid has a smell, but it doesn't feel out of place in most sweets.