GesselViennot's simple but powerful lemma has many striking applications, such as counting noninsecting paths , proving the JacobTrudi's identities, and solving the aztec diamond problem. So I wonder wether it can also be used to prove the Dodgson's condensation and Newton's indentities. Also, if you know other theorems or identities that can be solved by this lemma, please let me konw...
I think that Viewing determinants as nonintersecting lattice paths yields classical determinantal identities bijectively by Markus Fulmek is your friend. 


A little offtopic: GesselViennot's lemma was also discovered by Karlin and McGregor (1959, "Coincidence Probabilities") and was used to construct dynamics of noncolliding systems of particles: take N identical particles evolving independently (e.g., under a onedimensional diffusion) and impose the condition that their trajectories do not intersect. Under some conditions you will get new interesting Markov particle dynamics with determinantal formulas for the semigroup and for the dynamical correlation functions, see also Koenig arXiv:math/0403090. Concerning the original question: GesselViennot's lemma provides a powerful combinatorial formalism to produce totally positive matrices, and in the case of Toeplitz matrices any such totally positive matrix can be obtained this way. So this could possibly lead to the proof of the facts that you need. 

