Could a complete mapping of the neural network be accomplished?
The C elegans brain has been completely mapped in terms of connectivity a long time ago. See for instance:
Would it be possible to artificially trigger a neuron to verify the mapping?
Interesting thought - I'm not aware of anyone having done that but it's been a while since I followed the C elegans literature closely.
C elegans is pressurised, so you can't easily stick an electrode (assuming you had one small enough) next to your neuron of choice to stimulate it. Maybe you could make a secific neuron become light-sensistive and use optical stimulation (the worm's transparent, so that helps) but again, not sure anyone has ever attempted that in C elegans.
The converse has been done repeatedly though: either ablate a neuron using a laser or design a mutant that won't have it in the first place (remember that the C elegans genome is completely mapped too) and see how it affects behaviour.