The matter wave is just a probability amplitude, a wave function representing for instance the probability of localizing an atom, molecule etc in space. Note that electrons are wave matters as well.
if two such bodies interact, what you describe as they are "going through theirselves" of course their "paths" are modified.
To be correct you have to sum over all possibilities which have their probability amplitudes and the result will be also in a form of superposition.
i suggest to read more about quantum mechanics description of scattering.
I seriously doubt if any matter waves of other electrons tunel into the wire. Instead you may expect simple classical electromagnetic interactions related to capacity, impedance, induction etc. I think Maxwell eqs. will be just enough unless you do experiments at nano scale, like in atomic force microscope where tunneling effects are crucial.
An electron passing through a neutral atom matter wave (e.g,. prepared Bose-Einstein Condensate dilute gas) would do nothing measurable. There might be a single, classical interaction, i.e., possibly a loss of one condensate atom.
In Quantum physics we have a principle which is called " Complementary principle".
according to this principle : All the "systems " or "mechanisms " like the electrons can be waves or particles, and this is relating to the conditions that has been prepared for them . although this principle is being used for microscopic particles but we can use it for larger mechanisms too, however we should note to some situations .
now, in wavelike mediums electrons can behave by their wavelike features, so because of being wave they can be in one line or different paths and IT IS relating to their "interacting conditions" besides depending on "probability" according to Quantum, thus we are never ever able to say, they will changes their path or not by "certainty ".