- "Firstly, QED is not exact. At a certain energy scale, we have electroweak symmetry breaking. Thus, many physicists don't care about predictions of pure QED at high energies, as it is not physical."
Answer: The modern point of view is, to regard QED (to be concrete, let us think of the electromagnetic interaction of the electron/positron field) as an "effective quantum field theory", valid only below a certain energy scale, where contributions from other degrees of freedom (muon, hadrons, etc.) described by the standard model become relevant. By the way, this "philosophy" applies also to the standard model itself, which can be seen as a "low-energy effective theory" valid up to an (yet unknown) energy scale, where "new" physics might become relevant.
- "... we can study pure QED mathematically."
Answer: Sure, enjoy yourself!
- "QED is perturbatively renormalizable ..."
Answer: Yes, the QED Lagrangian is a gauge theory, containg only terms up to (operator) dimension $4$, being thus (perturbatively) renormalizable. This means, that only the electron mass $m_e$ and the fine-structure constant $\alpha \simeq 1/137$ are needed as experimental input for pure QED (of $e^\pm$ and the photon), all other observables (like the anomalous magnetic moment of the electron) can then be predicted.
- "... perturbative expansions in QED are only asymptotic series. One expects that maximum accuracy is gotten by truncating the sum at $\sim 1/ \alpha \sim 137$, and that adding more terms will lead the series to diverge."
Answer: Perturbation theory in QED leads indeed to an asymptotic expansion in the coupling parameter. The divergence of the perturbative series of pure QED has no physical relevance. In calculations of the contributions of pure QED to the anomalous magnetic moment of the electron and the muon, the perturbative expansion has been extended up to $5$-loop order, meeting the current experimental uncertainties of these observables.
- "... So naive divergence of the perturbation series does not necessarily mean the theory is ill-behaved."
Answer: This might be true or not, but this question is completely irrelevant from a physical point of view. The applicability of pure QED breaks down for purely physical reasons (presence of other particles) well below those energy scales, where this problem might appear (see also 6.).
- "There seems to be a Landau pole, but apperently this can't be trusted as the calculation leading to this prediction is perturbative."
Answer: The energy scale (the Landau pole in ancient terminology) $\Lambda_{\rm QED} \simeq m_e \exp(3 \pi/2 \alpha) \simeq 10^{277} \, \rm GeV$, where the perturbative expansion breaks down, is beyond good and evil. Compare it to the Planck mass $\simeq 10^{19} \, \rm GeV$, where gravitation becomes relevant! Seen as an effective low-energy theory, pure QED of the $e^\pm$ field is valid only up to an energy scale, where the presence of the muon ($m_\mu \sim 100 \, \rm MeV$), etc. becomes relevant. Seen as the unbroken $U(1)$ part of the standard model (taking into account all presently known particles), its applicability might be up to $\rm TeV$ energies.
- "QED being trivial..."
Answer: Triviality of a theory simply means, that taking such type of a theory seriously up to arbitrarily high energies (or, equivalently, down to arbitrarily small distances), i.e. requiring all the "axioms" of a relativistic quantum field theory, it only admits the free theory solution. Regarding QED as an effective theory, this "problem" is again irrelevant from a physical point of view.
