(Non)Existence of limits

Another example is $$\lim_{x\to 0}\frac1{x^2}$$ which does not exist in $\mathbb R$, but does exist in the extended reals $[-\infty,\infty]$.

It also exists in the extended complex numbers (Riemann sphere, roughly $\hat{\mathbb C}=\mathbb C\cup\{\infty\}$).

Note that $$\lim_{x\to 0}\frac1{x}$$ does not exist in the reals or the extended reals, but does exist in the Riemann sphere. That's because there is only one infinity in the Riemann sphere, but not so in the extended reals.

Yes for example a sequence of rational numbers might converge to $\sqrt{2}$ which is not rational. . So the sequence does not converge in $\Bbb Q$ but does converge in $\Bbb R$.

If your sequence is in $\Bbb R$ though, then since $\Bbb R$ is complete, if the sequence is Cauchy then it does converge somewhere. So if it doesn't converge in $\Bbb R$ it won't converge anywhere bigger because it won't be Cauchy.

For example the sequence $a_n = \bigg(1+\frac{1}{n}\bigg)^n$ has all $a_n \in \mathbb{Q}$ but $$\lim_{n\rightarrow \infty} \bigg(1+\frac{1}{n}\bigg)^n =e \notin \mathbb{Q} $$

If however, $a_n \in \mathbb{R}$ then $(\lim_{n\rightarrow \infty} a_n) \in \mathbb{R}$ (if it exists) since $\mathbb{R}$is complete.