How do I determine the temperature, so that I may calculate the power needed for melting i.e. get 100 centigrades? I assume I have to use the temperature coefficient and wire diameter (0.15mm)?

The heat output is where you will run into difficulty. Heat is lost from the wire by convection (air currents), conduction (wire contact with the material being melted), and radiation. The methods for calculating these kinds of heat loss have so many uncertainties that it is much better to do a practical experiment instead. Connect your wire to an adjustable power supply and slowly increase the voltage until you get the desired cutting/melting performance.

Ian is correct, and the equations are complex, usually involving calculus. And the power losses are your enemy..if you are trying to cut ice in a snowstorm in the arctic with such a hot wire, then it will need much, much more power (heat input) then if you are cutting foam on a still day in the Sahara Desert.

As Ian said, it's often better just to try it and see emperically. Sometimes, however, it is useful to have at least a rough idea if you need a 10W PSU or a 1000W PSU..

So...

how much heat input (power) will be required to raise the wire to 100C, assuming no cooling and before doing any work with it (work will take energy out of the system)? This is basic physics 101

This will give you a heat input requirement, in Joules. Remember 1 Joule is 1 Watt in 1 second, or a watt-second. You can use any combination of V x A for n seconds to get the Joules you need, as long as your wire can handle it. Of course you need to keep inputting this energy because of the losses mentioned already. If you exceed the Volts or Amp rating of your wire it will go poof

If you don't put in more energy than your losses, it will never heat up. If you can't get the Joules you need without exceeding some maximum of your wire (i.e. Volts or Amp ratings), then you will never get to that temperature., and you need a different wire. In this case your probably OK, since 100C is easy for a resistance wire like constantan.

I took the density and specific heat capacity, S, of constantan from Wikipedia. r is the radius of the wire, and the area is thus the cross sectional area. delta-T is the temperature rise above ambient. So if you want 100C, you want to raise it by 75C, assuming ambient is 25C.

Because of the power losses you will need to input more energy then this equation indicates. But its a starting point. There will be thermal transfer to the surrounding air, so you need to input more energy to account for that... and when you go to use it, it will cool down because of conductive losses into the material being cut, so you need more energy for that loss as well; cutting a block of ice is different than cutting foam, its clear to see that cutting ice will require more energy input to the system.