complex numbers - Explanation of Euler's identity

I am not math student and I don't do math professionally, so sorry for the stupid question (if it is such). I'm interested in Euler's identity:

$$e^{i \pi} + 1 = 0$$
or
$$e^{i \pi} = -1.$$

Is it the same as like $\qquad e^{ix}=\cos(x) + i\sin(x)$ ?

Can you give some physics or real life examples of using it and what's the difference between the two formulas.

I understand the idea behind $e^x$, simply explained it is the amount of continuous growth after a certain amount of time($x$). But why we use $\pi$ or $i$, in which cases and what they represent?

In which cases we multiply $e$ and in which we put it to the power of another number(difference between $e^x$ and $ye^x$)?

Some articles, books or videos will be helpful. I watched some, but they only explain how to prove the equation and how to find the derivative, not the using of it.