All metals expand with heat so the nichrome wire at cutting temperature also expands and gets longer. Because of this, some method of keeping tension on the wire is needed in hot wire foam cutters. This is usually accomplished either with a springy frame that the wire is stretched between or a spring is used. It is also conceivable that a weight could be used with the wire over a pulley. Tension on the wire also helps keep it taunt so when a little pressure is applied when cutting foam, the wire stays fairly straight which is necessary for a good quality and uniform cut.
Because of the need for tension, the smaller the wire is, the less tension can be applied without breaking or permanently stretching the wire. Using 40 gauge wire means very little tension is possible and it will be harder to keep the wire taunt when cutting. The longer the wire is, the more pressure needs to be applied to the wire to keep it straight and taunt. This is why as a general rule, the longer the wire, the thicker it needs to be. There is no length vs. gauge size standard since theoretically any size can be used at any length with the proper voltage applied and current capacity of the power supply.
The temperature of a straight wire in room temperature calm air can be calculated. A given temperature will result in a specific current flowing through a specific diameter wire. It doesn't matter how long the wire is, a given current flowing through the wire will result in the same temperature. For example, a 26 gauge wire with 2.1 amps flowing through it will result in 600 degrees F whether it is 2" long or 200" long.
The bigger the diameter, the more current is required to heat it to the same temperature. For example, only 0.31 amps will result in 600F in 40 gauge wire, but 11.6 amps is required for 14 gage wire. In addition, the larger the diameter wire, the longer it will take to reach the equilibrium temperature.
The reason a strait wire reaches a given temperature and stays there in calm room temperature air is that the current continues to produce more heat as long as the current flows. At the same time, heat is being transferred away from the wire to the surrounding air. The hotter the wire is, the faster the heat is transferred away. The wire reaches its equilibrium temperature when the heat generated is equal to the heat transferred away.
If you coil the wire, in a tight coil like in heaters, transfer of heat away from the wire is reduced because there is more wire in a given volume of air and so the wire will get hotter.
In the same way, a wire in contact with any other material will change the rate of heat transfer away from the wire. If the material it is in contact with is a good conductor of heat such as copper, the equilibrium temperature will be lower because heat is transferred away faster. If the material it is in contact with is a poor conductor of heat (an insulator) the equilibrium temperature will be higher because less heat is transferred away. These situations result in complicated heat transfer equations that are not easily solved. In this case, experimentation is required to find the right wire and voltage to create the desired temperature.