Solution:

A domino $(i, j)$ is an oriented edge of the complete graph on vertices $1, \ldots, 40,$ and the rule that $(i, j)$ and $(j, i)$ cannot both appear means each of the $\binom{40}{2} = 780$ edges is available at most once. A proper sequence is exactly a trail: a walk that repeats no edge. In any trail, every vertex other than the two endpoints is entered and left equally often, so it has even degree in the set of edges used.

In the complete graph every vertex has odd degree $39,$ so at least $38$ vertices must have odd degree in the set of unused edges, and a graph with $38$ odd-degree vertices has at least $\frac{38}{2} = 19$ edges. Hence at most $780 - 19 = 761$ dominos can be used.

Conversely, set aside the $19$ disjoint edges $(3,4),$ $(5,6),$ $\ldots,$ $(39,40).$ The remaining graph is connected and only vertices $1$ and $2$ have odd degree, so it has an Euler trail traversing all $761$ remaining edges; orienting each edge in the direction of travel gives a proper sequence of length $761.$