What does algebraic number look like locally? Is there any theorem characterizing what algebraic number looks like locally (in completion)? For example, do all algebraic numbers live in some $\mathbb{Q}_p$? Does there exist algebraic number in extension of $\mathbb{Q}_p$ of degree p, say?
Let me formulate your question (or rather one of your questions) this way: if $K=\mathbb Q(\alpha)\supset\mathbb Q$ is a finite extension, is there an embedding $K\to\mathbb Q_p$ for some $p$? The answer is yes; supposing without loss of generality that $\alpha$ is an algebraic integer, if $f\in\mathbb Z[x]$ is the monic minimal polynomial of $\alpha$, there are infinitely many $p$'s such that the reduction of $f$ mod $p$ has a root in $\mathbb F_p$ (one can use the nuke called Chebotarev's density theorem to see it, but I guess there are simpler proofs). Take such a $p$ not dividing the discriminant of $f$, and by Hensel lemma $f$ will have a root in $\mathbb Q_p$.
$x^2 + 1 = 0$ does not have a solution in $\mathbb{Q}_3$ since $0^2 + 1 = 1$ and  $1^2 + 1 = 2^2 + 1 = 2 \text{ mod } 3$.