Injection of $E(\mathbb{Q})_{\text{tors}}$ into $\tilde{E}(\mathbb{F}_p)$?

Let $K$ be a discretely valued field whose residue field is perfect of characteristic $p$, and let $E$ be an elliptic curve over $K$ with good reduction. If one supposes further that the absolute ramification index $e$ of $K$ is $< p-1$, then it follows that the reduction map is injective on the $K$-rational $p$-power torsion points of $E$. As you already noted, the reduction map is automatically injective on the prime-to-$p$ torsion, and so we conclude that (when $e < p-1$) the reduction map is injective on all the $K$-rational torsion.

There are various ways to see the claimed injectivity. One is via a consideration of the formal group, and another is via the theory of finite flat group schemes. The rough idea is that any non-trivial $p$-torsion point which reduces to the identity (and hence lies in the points of the formal group) is the solution to an Eisenstein polynomial of degree divisible by $p-1$. If $e < p-1$ then such a polynomial can have no roots in $K$, and so the non-trivial $K$-rational $p$-torsion points cannot lie in the formal group.

If one considers the particular case when $K = \mathbb Q$ or $\mathbb Q_p$ equipped with the $p$-adic valuation, where $p$ is an odd prime, then $e = 1 < p-1$, and this result applies. This is what Silverman is using.

I think this is discussed somewhere in Silverman, although I forget whether it is treated in the general form described above, or just in the particular case of odd primes $p$ in $\mathbb Q$.

I don't know if there is something else, but just from looking at what you quote, I would argue that the reduction modulo $3$ tells you that there can be no $5$-torsion (in fact, that there is only $2$-torsion and perhaps some $3$-torsion). Then the reduction modulo $5$ tells that there is some $2$-torsion, and perhaps some $5$ torsion. Putting the two together, once concludes that there is neither any $3$-torsion, nor any $5$-torsion, so that the reduction maps actually provide embeddings of the torsion group, and the argument proceeds from there.

Basically, if reductions modulo $p$ and modulo $q$ reveal only $\ell$-torsion, with $p$, $q$, and $\ell$ pairwise distinct primes, then you know that there can be only $\ell$-torsion and that the two reduction maps are in fact embeddings.