  Given a class of differential equations with arbitrary element, the problems
of symmetry group, nonclassical symmetry and conservation law classifications
are to determine for each member the structure of its Lie symmetry group,
conditional symmetry and conservation law under some proper equivalence
transformations groups. In this paper, an extensive investigation of these
three aspects is carried out for the class of variable coefficient
(1+1)-dimensional nonlinear telegraph equations with coefficients depending on
the space variable. The usual equivalence group and the extended one including
transformations which are nonlocal with respect to arbitrary elements are first
constructed. Then using the technique of variable gauges of arbitrary elements
under equivalence transformations, we restrict ourselves to the symmetry group
classifications for the equations with two different gauges g=1 and g=h. In
order to get the ultimate classification, the method of furcate split is also
used and consequently a number of new interesting nonlinear invariant models
which have non-trivial invariance algebra are obtained. As an application,
exact solutions for some equations which are singled out from the
classification results are constructed by the classical Lie reduction. The
classification of nonclassical symmetries for the classes of differential
equations with gauge g=1 is discussed within the framework of singular
reduction operator. Using the direct method, we also carry out two
classifications of local conservation laws up to equivalence relations
generated by both usual and extended equivalence groups. Equivalence with
respect to these groups and correct choice of gauge coefficients of equations
play the major role for simple and clear formulation of the final results.
