How do I substitute proofed dried active yeast for fresh yeast?

In both cases, you don't add the recipe's amount of water to the proofed yeast. If your recipe says e.g. 500 g flour, 300 g water and 10 g fresh yeast, you measure these 300 ml water, then pour some of the 300 ml over 10 g of pressed yeast to proof it, adding a teaspoon of sugar if you want it quicker. After that, you mix flour, proofed yeast and the remaining water together (for simplicity, I left out salt and possible other ingredients).

You do it exactly the same way with dry yeast, only you have to use the correct substitute ratio, which is 3:1. So you measure 300 ml of water and (1/3)*10 g = 3.3 g of dry yeast. Then you pour some of the 300 g of water over the dry yeast, and after it has bloomed, you mix flour, sponge and the remaining water. The bread hydration stays correct, and the fermentation time/amount is equivalent to the fresh yeast case.

Proof the yeast in the water you mention in the last sentence.

If you want to make sure it works properly, use instant and not active dry. Active dry will probably work, but you can't really rely on it. Instant yeast is a little closer to fresh yeast and keeps a lot more reliably. I have heard that on occasion a recipe pops up that requires dissolving the yeast in cold water; for that, you want cake yeast and nothing else, because dry yeasts can't handle the cold shock. Other than that, cake yeast is kind of a pain and not usually worth the effort.