How is a Twisted Pair's +-2.5v used to send a 1 or a 0?
If the 100BASE-TX runs +-2.5V, does that mean the TX+ wire uses 0 through +2.5V, and the TX- wire uses -2.5V through 0?


Not usually.


  What is the voltage range that equates to 1, and what is the voltage range that equates to 0?


Usually, you'd transmit a 0 with, for example TX+ = -2.5 V and TX- = +2.5 V. And you'd transmit a 1 with TX+ = +2.5V and TX- = -2.5V. 

While, as Andy says, in theory it isn't important if the two signal lines have different bias points, in practice it's easiest to make a receiver that compares the two lines to each other. So that any situation where RX+ > RX- is interpreted as a one and any situation where RX+ &lt; RX- is received as a zero.


  Who and how is this range set?


As I said above, any situation where RX+ > RX- is interpreted as a one and any situation where RX+ &lt; RX- is received as a zero. 

The specifications for the allowed common mode voltage and signal amplitude are usually chosen based on the available technology (semiconductor process and known circuit designs) so that all users can design their circuits to be interoperable. But basically this is an arbitrary choice made for different standards (LVDS, Ethernet, etc.) based on the technology expected to be used to implement them.


  How is the noise removed?


Noise is removed because the receiver is designed to be mostly sensitive to only the difference between the input voltages, and have as little as possible response to the common mode voltage of the inputs. 


  If 100BASE-TX uses +-2.5V, in a typical office environment, about how much mV of noise can you expect?


You can see how much noise and interference the system designers allowed for by looking at the minimum signal amplitude allowed at the receiver. To give very low error rates, the maximum noise allowed is probably 1/15 or less of the signal amplitude.