Approximation Lemma in Serre's Local Fields

As for "by linearity": if you want to find $x$ that is $p$-adically close to $x_1$ and $q$-adically close to $x_2$, then you can do this as follows: suppose you can find a $y$ that is $p$-adically close to $x_1$ and $q$-adically close to $0$, and a $z$ which is $p$-adically close to 0 and $q$-adically close to $x_2$; then set $x = y + z$. That (or the $n$-variable version) is all that's meant.

For your other question, you can force an element not to be in an ideal by making it congruent to 1 modulo that ideal, for instance.

I agree with you that this is a (very rare!) slightly obscure moment in Serre's exposition.

This result however is found in absolutely every treatment of valuation theory / local fieds. It is a very famous and useful result, usually called either Weak Approximation or Artin-Whaples Approximation.

For instance, I give (what I want to be) a quite careful, detailed proof of this result in $\S 1.4$ of these notes. I took the proof from a classic text of Artin, so it's a good bet that it is the original Artin-Whaples proof. Also, this proof is "constructive".

Note that what I (and most others) call Artin-Whaples approximation is slightly more general than the result you quoted above in that it concerns norms rather than (discrete) valuations: in particular, Archimedean norms are allowed. This extra generality doesn't make the proof any harder but really does come up in algebraic number theory: e.g. I have often needed an element $x$ of a number field which jumps through various $v_{\mathfrak{p}}$-adic hoops (for a finite set of prime ideals of the ring of integers) and is positive with respect to every real Archimedean place.