Does there exist a totally disconnected topological space T whose topology is the order topology of a linearly ordered set and whose (small inductive) dimension is equal to 1? There exist topological spaces which satisfy all the other conditions, but I do not know of any whose topology is an order topology.
No. According to this paper zero-dimensionality (in small, or large inductive, or covering definition) is equivalent to being totally disconnected, and all non-such ordered spaces are one-dimensional (in all three definitions). Dimension theory is pretty dull in ordered spaces.