Hi,
I don't think that even Unity expose an Api to know when a value is sent over the network. Is it not a case where you should use "ReliableDeltaCompressed" for network synchronisation, and smooth out the result using interpolation?
In the photon demo, the animation trigger system would be a good starting point if you want to do something without reliable sync. instead of only relying on the position, also rely on the user input itself, and check for input changes, when the user input changes, catch up to the current position, else make a blend between your prediction AND the network position value. Making a interpolation between the two will give you a smooth result I guess.
as for initial velocity, is it extrem velocity? is the player actually not moving at all, and when instancicated on other computer, it gets a velocity anyway? to solve this, you could force the velocity to 0 the first frame after the instanciation.
bye,
Jean