"the function ρ(m) is a process-dependent or so. Is it right?''
No. ρ(m) is characteristic of the infraparticle field itself, it figures in the field commutation rules, and hence appears in every state, including states of rest. m0 is the lower limit (branch point) of the mass spectrum, and is the substitute for the rest mass.
The infraparticle field gets its electromagnetic structure from nontrivial commutation rules with the e/m field. Because of gauge invariance a physical asymptotic infraparticle state automatically contains a coherent electromagnetic field - the e/m field created by the electron. In terms of photons, the coherent electromagnetic field consists (like any coherent state) of an indefinite number of soft virtual photons. There are no real photons involved as long as the asymptotic conditions prevail; real photons have their own asymptotic states.
For the electron branch cut and its relations to anomalous exponents see:
T. Appelquist and J. Carazzone, Infrared singularities and massive fields, Phys. Rev. D11 (1975), 2856--2861.