Does the orientation of transmit antenna affect the path-loss of a signal transmitted by an omni-directional antenna? If so, how are antennas optimally oriented? I am specifically interested in knowing how eNode antennas in LTE are oriented in the downlink transmissions to ensure propagation loss is minimized?
Yes, it does, in general.
Considering a simple half-wave dipole, which is what you're probably thinking of as an omnidirectional antenna, its radiation (or reception) pattern is strongest perpendicular to the elements, and has nulls off both ends. Additionally, the polarisation of the signal is along the elements. Since a dipole is typically mounted vertically (so it is omnidirectional with respect to the horizon), it will transmit (or receive most efficiently) vertically polarised signals.
A loop antenna can be mounted horizontally to give a horizon-omnidirectional pattern with horizontal polarisation. This will have nulls perpendicular to its plane.
A pair of cross-mounted loop antennas with a phase-shift network can be used to give circular polarisation in two specific (opposite) directions, and linear polarisation in other directions. Some people use these for contacting satellites without the need for pointing equipment.
Antenna theory is a very broad field. You should definitely choose an antenna type and mounting position to properly suit your application.