Does a proton have a binding energy?

Note that binding energy is relative, like the gravitational potential energy in certain respects. That said, the binding energy of the proton could be considered non-zero if its sub-particles, i.e. quarks, are studied. However, since in the energy region of your mentioned reaction these sub-particles play no direct role (except for perhaps the underlying physics), it is more reasonable if you consider the binding energy of the nucleons, i.e. protons and neutrons, as zero. Note that if you consider the binding energy to be non-zero for both sides of the reaction, similar final results would be obtained.

No, a proton doesn't have a well-defined binding energy. That would be the energy required in order to separate it into three quarks, but free quarks don't exist.

Yes. Protons aren't elementary particles, so they do have binding energy. What that means exactly is a bit trickier, since the structure of the proton isn't something we have a full consensus on yet.

There's two main ways of seeing protons:

• Protons are made out of three quarks. Each of these quarks only has a tiny fraction of the mass of the proton, and the difference is the binding energy. Note that the binding energy is positive, and huge - almost all of the mass around you comes from this binding energy.
• Protons are made out of a sea of quarks and their corresponding anti-quarks, except for three unpaired quarks. The mass of the proton comes from all of these quarks and anti-quarks, but since all except three are paired, most of the behaviour of the proton comes from the three unpaired quarks. But even in this case, the proton does have binding energy - it just doesn't have to account for 99% of the mass of the proton.

From UMD:

What one has learn about the nucleon structure through high-energy scattering? First of all, one learn there are indeed 2 up valence quarks and 1 down quark, with electric charge 2/3 and -1/3 of the proton, respectively.

(the properties of a proton are largely determined by the three valence (unpaired) quarks)

Second, the number of quarks is infinite because the integration does not seem to converge. This is because there are infinite number of quark and antiquark pairs in the proton.

Note that both are essentially the same as far as QFT is concerned - the massive binding energy would mean the spontaneous creation of quark-antiquark pairs; quark confinement explains why we can't ever observe these in isolation (the binding energy of quarks increases with distance, and eventually gets large enough that instead of more separation, new quarks are created). They're also essentially the same for outside observation - the mass of the proton is the same regardless of whether it comes from the binding energy of quarks inside the proton or the sum of the masses of the constituent quarks.