What quantity of baking powder satisfies a mass of size x?

About 5g baking powder per 150g flour, or about 1tsp of baking powder per US Cup of flour (Source: Michael Ruhlman's Ratio app.) That will work for most chemically leavened batters, quick breads, pancakes, etc. This is a guideline, not a rule. Sometimes more leavening is desirable— you might want a slightly puffier quick bread and you're using a dough with enough protein to handle it. Maybe you want less because you've got a batter that doesn't stay together on something you're frying.

Unless you're planning on doing some workshopping, I recommend using an existing proven recipe to get a good ratio of ingredients. After a while, you'll get a better idea of how much leeway you've got with each variable in the technique.

There is no such amount. Each recipe behaves differently, and a lot of experimentation is needed to finetune the ratios of all ingredients, including the baking powder. Change an ingredient, and you have to start all over again. Formulas are impossible here - you can use "average" ones for a given type of cake, like 1:1:1:1 for pound cake, but as soon as you start to change the base ingredients or adding other ingredients or change the mixing method, the way the cake leavens changes too, and you are likely to get into a situation where a different amount of baking powder is needed. And then there is personal or cultural preferences - what kind of internal structure is the goal.

If you are in a pinch, you can use some rule of thumb. Just be aware that it will not result in a good cake all the time, and almost never will be the optimal for a given recipe.

You already see this arbitrarity in the two answers you got, one is 5 g per 240 g flour and the other is 5 g per 150 g of flour. For what it is worth, European producers sell baking powder in sachets of 15 g, and usually print something like "for 500 g flour" on them, so 5 g per 167 g flour. However, I personally never use this ratio, because it tastes too metallic to me. But if you want to start working with a single number, you can pick any of the three, or another in that range, it doesn't really matter.

Well, it depends on the type of cake, so let's take a simple one.

One of the most basic cake batters there is is the standard 1-2-3-4 Yellow Cake. In this formula, the "3" stands for "3 cups self-rising flour", which is contains roughly 4.5tsp baking powder (and recipes which use regular flour bear this out).

So your mass here is:

• roughly 8oz Butter
• 2 cups sugar (14oz)
• 3 cups flour (14oz)
• 4 eggs (9oz)
• 1 cup buttermilk (8oz)

Totalling 53oz of ingredients to 4.5tsp baking powder, or 1tsp baking powder for every 12oz of cake batter, roughly. Or, in real measurements, 5g of baking powder for every 335g of batter.

However, that's a limited use ratio because cake recipes scale up and down poorly; both very large and very small cakes usually have to be adjusted to avoid over or under cooking, rounding too much, or center collapse.