Assuming deltaX=10 fermi; get Delta P(momentum) and find Kinetic energy =p^2/2*mass) , The energy of protons inside nucleus is of the order of 1-2 (few)MeV; thats also true. So they exist inside nucleus.

For electrons, it will come to be high say 20 MeV or so, but practically we never her such high energy beta particles in decay, so do not exist.

For protons case, energy obtained in that way, you describe satisfy vc. Can I add this argue to my answer?

No, NO. Please note that maximum v=c; which is limitation of classical Newtonian mechanics.

Why dont you try to find velocity of electrons having kinetic energy of 10eV, 1 keV and 1MeV, 10 MeV and so on using both classical and relativistic methods. At low energy both will give same answer whereas at higher K. energy; only relativistic holds.