Marking and measuring the intervals will introduce random error. Yes.
The pressing of the probe is more tricky. If you assume that the amount of pressure you use each time is random, then yes.
I'm not sure what the significance of 0V is without seeing a circuit diagram, showing how you have connected the supply to the resistance paper, and the meter to this.
If you are trying to find the point where the thickness changes then it's where the two straight lines cross on the graph. As you are measuring changes in voltage along the paper, it's not really the actual value of the voltage at any point that matters, but the change in voltage from one point to the next.
For example, if you measured -0.8V, -0.5V, -0.2V, +0.1V, +0.4V along the paper at equal intervals, it would have exactly the same significance (a straight line) as +0.3V, +0.6V, +0.9V, +1.2V etc.
The zero volt level in a circuit is usually "earth potential" and has no significance other than that it is an arbitrary reference voltage. I'm not sure why this should be in the middle of the paper somewhere. As I say, without the circuit details I really can't say.