Please help me to find the sum $\sum\limits_{n\geq1} \frac{\sin(nx)}{n^2}$

As you observed, by the Weierstrass M-test, the series converges uniformly. Let $$f(x)=\sum_1^\infty \frac{\sin(nx)}{n^2}.$$ The functions $\sin(nx)/n^2$ are continuous, so their "sum" $f(x)$ is continuous. (In general, if we do not have uniform convergence, but only pointwise convergence, the sum need not be continuous.)

The above result about continuity is causing you some confusion. It is based on a theorem that says, or should say, that if $(f_n(x))$ is a sequence of continuous functions that converges uniformly to $f(x)$ in an interval, then $f(x)$ is continuous.

To apply the theorem to a series $\sum_{n \ge 1} g_n(x)$, we just let $f_n(x)=\sum_{k=1}^n g_k(x)$. There is nothing new in this: convergence of infinite series is defined in terms of what happens to partial sums as $n$ goes to infinity.

The additional fact that is needed is that a series uniformly convergent on a finite interval can be integrated term by term on that interval. So we need to calculate $\int_0^\pi \frac{\sin(nx)}{n^2} dx$ and "add up."

More explicitly, $$\int_0^\pi \left(\sum_{n\ge 1} \frac{\sin(nx)}{n^2}\right)\,dx=\sum_{n \ge 1} \left(\int_0^\pi \frac{\sin(nx)}{n^2}\,dx\right).$$ (The above interchange of the order of integration and summation is permitted because of the uniform convergence. It can fail when we do not have uniform convergence.)

The integration is straightforward, for $\sin(nx)$ has $-\dfrac{\cos(nx)}{n}$ as an antiderivative.
Thus $$\int_0^\pi \frac{\sin nx}{n^2}dx=\frac{1}{n^3}(-\cos(n\pi)+1).$$ If $n$ is even, $\cos(n\pi)=1$, so the integral is $0$, and contributes nothing to the ultimate sum.

If $n$ is odd, then $\cos(n\pi)=-1$, so the integral is $\dfrac{2}{n^3}$. Since $n$ is odd, let $n=2m-1$. Then $$\int_0^\pi \frac{\sin nx}{n^2}dx=\frac{2}{(2m-1)^3}.$$ We conclude that $$\int_0^\pi f(x) \;dx=\sum_1^\infty \frac{2}{(2m-1)^3}.$$ (In your post, the final variable of summation is $n$, but that makes no difference.)

Interesting fact: The sum that we end up with is closely related to $\sum_{n \ge 1} \frac{1}{n^3}$. This last sum, also known as $\zeta(3)$, was proved to be irrational by Apery about $30$ years ago, using an elementary (but not easy) argument. It was one of those rare mathematical results that even gets reported in the mainstream press.

Differentiate twice, get the series for $-\dfrac{1}{2 \tan\frac{x}{2}}$. So now integrate this twice... Of course that is not elementary, but still...