A question on temperature regulation of a nichrome wire
Is the hotwire's temperature proportional to the Iavg or Irms. 


\$ I_{RMS} \$ is the current which will give the equivelant heating to the same DC current so that's what you want.


  But if the temperature is directly proportional to Irms ...


Power is directly proportional to \$ I^2R \$. Nichrome has a positive temperature coefficient (check this is true) so R won't be constant.

Stable temperature will be the point at which electrical power in = power lost from the wire. If the latter is not constant then temperature won't be either.


  In my case, regulating the PWM current's average or rms value to a pre-set value should be used?


I would be inclined to regulate to wire resistance. That way you are controlling to a temperature. Measure both current and voltage, calculate resistance and control to that.


  Is temperature only related to current in this case? 


See above.


  Length has no significant effect? 


There'll be some additional cooling at the connectors. The shorter your heating wire the more significant these become.



From comments:


  If I could use a differential amplifier and measure the average voltage across the nichrome as well as the current. How would you formulate the temperature?


Take a reference reading after the power has been off for enough time to reach room temperature. Call this 100%. Measure the resistance thereafter, calculate the percentage increase in resistance and use a lookup table or linear approximation to calculate actual temperature.




  I'm kind of confused. Some comments under my question mention the current is proportional to the heat of the wire. See also this link: hotwirefoamcutterinfo.com/_NiChromeData.html It says in a paragraph: "Know that it is the CURRENT not the voltage nor the wattage that heats your wire. Rather, it is the actual passing of current through the wire that ultimately determines its temperature."


The article is a little short on correct technical explanation. The last line is confusing because V and I are related by R. What they are trying to get across is that controlling the current means that you get the same result (for a given wire gauge) independent of the length. If you were controlling by voltage you would need to adjust the voltage to suit the length of the wire.

This comment is the first mention of foam cutting. I suspected that might be your application and that's why I spoke of power. There are a few points to note:


The temperature, as already mentioned, will stabilise at the point where heat into the wire = heat lost. 
This will change depending on whether the wire is in air or cutting foam. 
It will change with your speed of cutting. The faster you cut the more heat power will be drawn from your wire and if you go too fast the wire will cool enough that it won't cut and you will break it.
The portion of wire outside the foam will be at a higher temperature than that doing the cutting since it is not loosing heat at the same rate.



  Rather, it is the actual passing of current through the wire that ultimately determines its temperature."


Again, the current sets the power. The temperature depends on what's happening outside the wire.



I think you shouldn't over-complicate your design. Make your power controller adjustable and add some sort of current indicator - maybe just your multimeter - in series with the wire. Find a current value that works for your foam cutting speed and write it down!