Edited after question was edited:
1 - Is it correct to say that, in the first order, each virtual photon changes the phase by a certain amount, and the total phase is then given by an average number of virtual photons absorbed? In the first order (tree level), the Feynman diagram is just a "Y" diagram, with the external field drawn as a (crossed) virtual photon being absorbed by the electron.
No, it is not.
Virtual photons are a mathematical construct of quantum field theory, they are integrated over in the final integral, they are not real countable photons. Example
The contribution of the virtual line is under an integral, with a propagator which has a continuum over the Q^2 of the integration. The continuum is not countable.
The virtual photon is not physical, it is just a mathematical representation of quantum numbers transferred and energy/momentum exchanges.
2 - How does the number of virtual photons arise in the first order description?
It does not.
Virtual photons are not a countable construct, they are on a continuum of energy and momentum values.
3 - What is the phase change due to one virtual photon in the first order? Feynman, in his book QED, writes about this phase change.
One virtual photon means a d(E) d(p) change under the integral . The contribution of the line is infinitessimal, controlled by the integration.
You have not given an accessible link to the book, here is the discussion of Feynman I have in mind. AFAIK the phases are discussed in the solutions of the basic quantum mechanical equations. These phases appear in the solutions of the appropriate quantum mechanical equation (the dirac in this case) and a phase is added in the wavefunction solution if a magnetic field is introduced.
The lowest order diagram will have all the phase change within the approximation of first order . Calculating higher orders will reduce the error in comparison with data ( though it is only recently that measurements claim to be able to measure the real and imaginary part of a wavefucntion, it is only the complex conjugate squared that is measurable afaik)