I was thinking about adding $0$ to an equation, e.g.:
A very simple one:
$$2x + 2 = 10\\
2x = 8 \\
x = 4 .$$
If you add "$+ 0$" to any side it does not change the outcome. $2x + 2 + 0 = 10 \implies x$ is still $4$.
Are there any equations, formulae or mathematical constructs where adding "$0$" does change the outcome?
When you have an equation, there are only two (types of) things you're allowed to to. 1: make a change to one side that does not change the value of that side. 2: do the same thing to both sides.
Adding $0$ to an expression does not change its value, and as such, you are allowed to do it to one side of an equation and not the other. Most of the time adding $0$ is just a gimmick, but some times it can be very valuable. Particularily in the form of "Add something, then subtract the same ting". One example I like is when proving the formula for the derivative of a product of two functions. There you have the following transition: (note, these are not equations, trying to solve for an unknown, but equalities, finding as simlpe as possible an expression for some predetermined value) $$ \frac{f(x+h)\cdot g(x+h) - f(x) \cdot g(x)}{h} \\ = \frac{f(x+h)\cdot g(x+h) \color{red}{-f(x+h)\cdot g(x) + f(x+h) \cdot g(x)} - f(x)\cdot g(x)}{h} \\ = \frac{f(x+h)\cdot\big[g(x + h) - g(x)\big] + g(x)\big[f(x + h) - f(x)\big]}{h} $$ where the fact that we add $-f(x+h)\cdot g(x) + f(x+h)\cdot g(x)$, which is zero, allows us to factor the numerator into something we can handle later on.
So, adding $0$ does not change the value of any expression, and thus is something you're allowed to do at any time. However, some times it can greatly affect the look of an expression, allowing for algebraic manipulations that were unavailable before.