Is my proof that $\frac{\pi}{4}=\sum\limits_{n\geq 0}(-1)^n \frac{1}{2n+1}$ correct?

You are already 90% of the way there; using term-by-term integration or any of the other methods described at Taylor series of $\arctan$, you can derive

$$\arctan x=\sum_{n=0}^\infty \frac{(-1)^nx^{2n+1}}{2n+1},\qquad |x|<1,$$

but you cannot claim directly that this is true for $x=1$ because this is out of the range of the equation. (You can't do term-by-term integration on the interval $[0,1]$ because the $\sum\int=\int\sum$ theorem is only valid when convergence is uniform, and a priori you only know that the power series is uniformly convergent on $[0,x]$ for any $|x|<1$.)

But with a little extra work, we can push this equality out to the boundary of the unit circle, using Abel's theorem. This says that if $\sum_{n=0}^\infty a_n$ is a convergent series then $$\lim_{x\to1^-}\sum_{n=0}^\infty a_nx^n=\sum_{n=0}^\infty a_n.$$

And we know that $\sum_{n=0}^\infty \frac{(-1)^n}{2n+1}$ is convergent, because it is an alternating series. Thus $$\sum_{n=0}^\infty\frac{(-1)^n}{2n+1}=\lim_{x\to1^-}\sum_{n=0}^\infty\frac{(-1)^nx^{2n+1}}{2n+1}=\lim_{x\to1^-}\arctan x$$ (because the power series is valid for $|x|<1$), and this is equal to $\arctan 1=\frac{\pi}4$ because arctan is continuous in a neighborhood of $1$ (indeed, in a neighborhood of $\Bbb R$).

By the way, just yesterday I submitted a formal proof of this exact theorem in Metamath, so I got to see all the little complications that are glossed over in this exposition.