A prototypical reflex arc just consist of two or three neurons, an afferent (sensory) neuron, possibly one or more interneuron(s), and an efferent (motor) neuron. A reflex arc is a rather direct connection, e.g. just from the sensory organ to the spinal chord and from there to the effector organ – it does not involve processing by the higher parts of the nervous system. To answer your second question: this is one path, consisting of two or more parts (two or more neurons).
It is conceivable that some manipulation would change the reaction time. If we could e.g. somehow replace one of the three neurons with a longer version, the reaction time would somehow grow, and since there are several neurons, their processing time might indeed be manipulated individually. It would not really matter which one of the neurons was manipulated, or if more than one would be manipulated. The total additional reaction time would be the sum of the individual additional reaction times.
To answer your first question: at which of the three neurons the signal were to take longer (»the timing would be different«) would depend upon which one of them was previously manipulated.
There are reflexes that involve more complex neural structure, but nothing like (conscious) bodily feelings/pain.
The withdrawal reflex that you mention is likewise based on receptors in the skin that through their relatively direct linkage with motor neurons can effect a withdrawal reaction without any (consciously) perceived heat or pain.