Should the period of the orbit depend on its orbital eccentricity?
Not on the eccentricity. A Keplarian orbit is defined by several parameters, usually seven. One parameter is for the gravity of the central body, two are for the orbit shape & size, three for how the orbit is pointing in space and the last parameter is for where the orbiting body is on the orbit at a given time. The period depends on the size (semimajor axis), not the eccentricity (shape).
