The spread in mass values you see in the plots comes from limitations in detector resolution. To determine the mass of the parent particle from its daughters, you need to measure the directions and energies of the outgoing leptons, and then determine the invariant mass of that system (in this...
That's right. This approximation is good to about 10% in the ratio of partial widths, as the ratio of phase space factors is ##\frac{(m_K^2-m_e^2)^2}{(m_K^2-m_\mu^2)^2}=1.1##. This is a much worse approximation to make for charge pion decay (ratio of phase space factors ##\sim5.6##.)
The primary absorber itself is a set of steel-encased, water-cooled carbon slugs, each about two feet across. Additional concrete and iron shielding surrounds this. When the beam is dumped, it is swept across the face of the carbon so that the load is spread around.
This isn't a general rule. Hadronic-only strangeness-changing processes must be weak, but hadronic-only strangeness-conserving processes needn't be strong. The example in the original post can proceed via strong, weak, or electromagnetic diagrams (albeit that does mean it would be called...
I agree that a more fleshed out context would be needed to give a reasonable muon/pion separation strategy.
In its simplest form, which could be appropriate for (say) a fixed-target test beam, you just place two thin but fast detectors some distance apart and let the particles pass through...
Models of new physics that introduce new particle species or new couplings (types of interactions beyond the familiar electromagnetic, weak, and strong) can lead to an observable rate of muon-to-electron conversion, and there are loads of such models in the wild. New particles could come about...