Using a Steel Wire as the Heating Element
1) There is no resistor to limit the current, shouldn't it blow up some fuse?  


Heating elements are resistors. By choosing an appropriate wire diameter, wire material and wire length, the resistance (R) can be tuned so that the heater will draw a given amount of amps (I) at a given voltage (V):  R = V/I. To calculate how much current (I) you need at a given voltage (V) to reach a certain amount of heat output (P) in Watts, you divide the target heat output by the voltage: I = P/V.  

For example, if you need 30 W of heat at a voltage of 12 V, your heater has to draw 30 W /12 V = 2.5 A in order to do so. Your heater should thus have a resistance of 12 V / 2.5 A = 4.8 Ω.


  2) If a steel wire of that thickness is used as the heating element, what current is required? 


There isn't any "required" amount of current like a LED would have for maximum brightness. A heating element (resistor) will simply draw as much current as its resistance allows at a given voltage.  

The resistance of the heating element (and consequently, the current drawn at a given voltage) doesn't just depend on the wire diameter, it's also dependent on the wire lenght, wire temperature and the wire material. For this answer I'll neglect taking the temperature into account in order to keep the answer as simple as possible. The lenght (l) required of a given wire is l =  D^2 * π * R / (ρ * 4), where D is the diameter of the wire, R is the desired resistance and ρ is the resistivity of the material (dependant on temperature). 

Continuing with the previous example, a 4.8 Ω heating element made of 0.5 mm diameter mild steel has a length of (0.0005 m)^2 * π * (4.8 Ω) / ((15*10^-8 Ωm) * 4) = 6.28 m. The heating element in the video is clearly much shorter than 6.28 m long; it is not made of steel.
That heating element was almost certainly made of a high resistivity metal alloy designed specifically for heating applications; nichrome wire. Plugging the same numbers into the above equation, but changing the resistivity to that of nichrome, gives us a much more plausible length of 0.75 m.  


  3) Is it possible to rely on the contact resistance as a way to limit the current?  


Again, a simple resistor doesn't need current limiting. No, you could not use contact resistance as a way to limit current, you can't set a switch to "half off".  

If you need to vary the heat output with e.g. a knob, use pulse-width modulation