When you get outside the nucleus and talk of particle interactions, the framework is particle physics interactions, and the model is Quantum Field Theory. QFT describes simply the scattering and decay interactions of particles.
These diagrams are iconal representation of the first order terms in a perturbations series, showing two body scatterings , and to read them, one has to decide which are the incoming and which the outgoing particles. The rule is if the arrow points against the direction of time, the antiparticle is implied. From bottom to top for the first diagram (from left to right for the second) : taking the incoming particles as the scattering of the proton and an antineutrino electron , quantum numbers allow a neutron to be produced together with an e+.
1.Does this idea of proton-to-neutron only apply to protons in the nucleus, free protons, or both?
If this is the case, would accelerating a proton fast enough (causing a gain 1.29 MeV of energy) cause it to convert into a proton?
Particles obey special relativity., and Feynman diagrams use the fourvectors of the particles. (The mass in "E=mc^2" has nothing to do with the invariant mass that characterizes all particles, in all inertial frames. It is not used in studying data of particle physics.) Interactions are described in the center of mass system of the particles, the numbers can be transformed to any inertial frame afterwards, with Lorenz transformations.
I'm familiar with the Cowan-Reines experiment and how antineutrinos were used to convert protons to neutrons. Are we limited only to antineutrinos to cause such a transformation? Or could any elementary particle (say, an electron) cause a proton to convert into a neutron, given that it was supplied enough energy?
Any elementary particle can end up into producing a neutron when scattering off a proton, in a complex diagram, BUT quantum number conservation and charge conservation have to hold. This means to get a positron to conserve charge from the proton, lepton number conservation needs an electron antineutrino; so it will always be there in the first order diagrams.