Ah! But of course, your lower bound is really by $\exp(-\int \log{\max(p,q)})$ and not merely by the final $\exp(-\int (|\log{p}| + |\log{q}|))$ of your answer... This is great and it answers all my issues - thanks very much! Just to point out my initial observation for small enough Mahler measures: $\| \log{|P|} - \log{|Q|} \|_{L^1} + m(P) + m(Q)$ turns out equal to none other than twice the Mahler measure of the bivariate asymmetric polynomial $wQ(z) - P(z)$, which is bounded away from zero by Breusch-Smyth (if $P/Q$ does not preserve the unit circle).

1) Oh, but of course $p(z)−q(z)$ is not analytic at $z=0$, and it was absolutely crucial that $r(z)=z^m(p(z)−q(z))$ was a polynomial in $z$ and not in $z,z−1$... I see. Ah well, but that only makes this proof more interesting! 2) I don't think this is nearly so easy: a positive constant lower bound by a positive function of $m(P)+m(Q)$ instead. But I do know it at least for small enough Mahler measures.

@StanleyYaoXiao: Your question should be infinitely harder than this one. I don't think there is much of a similarity between a class number and a Mordell-Weil rank.

Sorry, I realized that my comment sounds less than clear, so let me add that apart from the remark in the first sentence, I do regard there both $K$ and $E$ as varying. Basically I am saying that a version of Silverman's specialization theorem can be used $r$ times over an $r$-dimensional char. $0$ base $T$ assuming that there exists an elliptic scheme $A/T$ such that $A(T) / \mathrm{tors} \cong \mathbb{Z}^n$. From this point of view the problem is geometric, and I am guessing that an explicit such $A/T$ is given by $A_{1,n+1} / A_{1,n}$ (but any other construction would suffice).

I am not sure whether it is fair to say that the fixed number field case should admit a negative answer; of course that comes down to a famous unsolved problem. For the existence of $E/K$, an iterated application of Silverman's specialization theorem reduces us to proving that the group of sections of the elliptic scheme $A_{1,n+1} \to A_{1,n}$ has rank $n$, generated by the $n$ 'point doubling' sections $(x_1, x_2,\ldots,x_n) \mapsto (x_1,x_1;x_2,\ldots,x_n)$, etc. (Here $A_{g,n}$ is the space of $n$-pointed abelian varieties of dimension $g$). This should be true.

One outstanding question here is whether or not the discreteness should be uniform if we restrict to the subset of the integer polynomials $P \in \mathbb{Z}[X]$ with a bound $M(P) \leq T$ on their Mahler measure.

Really nice! Your inequality doesn't need that $P,Q$ have integer coefficients, but only that $\big| \|P\|_{L^2}^2 - \| Q\|_{L^2}^2 \big| \geq 1$ if it is not zero. Here is another consequence of applying this remark to $Q = 1$. For every complex polynomial $P = \sum_{i=0}^d a_iz^i$ of degree $d \geq 2$ and with non-zero free term, there is a positive constant lower bound on the Mignotte height $\widetilde{h}(P / \sqrt{|a_0a_d|}) := \int_{\mathrm{T}} \frac{\log^+ |P|}{\sqrt{|a_0a_d|}} > 0.1$ (that appears in the Amoroso-Mignotte refinement of the Erdos-Turan equidistribution theorem).