1. A photon isn't really a wave 'of' an EM field. It's a wave of some abstract field which doesn't directly correspond to anything classical. When you get a many-photon field, it can show the behavior we call a classical EM field.
An electron is a state of another abstract field. It's not a more compact version of a photon. It has properties (e.g. participation in the weak force) which photons lack.
2. It's rumored that the electron (like other particle fields0 picks up its mass from interaction with a hypothetical Higgs field. This is over my head. I'm not sure whether this fact, if true, would justify calling it a composite of two fields.
It is, to the best of our knowledge in current interpretations of quantum mechanics, entirely unnecessary to postulate a separate 'particle' nature to the electron or anything else. Whether or not it turns out to be necessary to postulate another process, beyond the standard wave equation, is open to some dispute. Even if the answer is yes, the output of that process is still describable in standard quantum wave language. The famous 'wave-particle duality' is a relic of early days when people struggled to picture the genuinely mysterious quantum behavior in classical terms.
3. Even the curvature of space should propagate in wavelike ways. If that curvature is not itself a quantum object, big problems arise because any classical particle or field in principle could be used to violate the uncertainty relations, unraveling the logical consistency of quantum mechanics.
Now we get to the easy questions.
4. The wave can't have a single well-defined wavelength, because then it would extend uniformly over all space.
5. There can't be sharp boundaries to the wave, because that would involve infinite energies. The wave trails off continuously for ever.
(published on 09/16/09)