Do resistance have to heat? Not a physicist.
Is there a way to build resistances that do not heat when opposing current?
More generally, is it necessary to waste energy to resistance?
If I cut a wire, there will be an almost infinite resistance (unless the tension is large enough to break the air), but no energy cost.
If I had a switch with a duty cycle of 50 pc, I would oppose the flow of current similarly to a resistance without heating, maybe. Is that an actual method for loss free resistance?
 A: Resistance is defined by it causing energy loss. If you expose free electrons (or generally electric charges) to an electric field, they will be accelerated continuously. In a resistor (i.e. a conductor), by contrast, charges move at a certain average velocity, called drift velocity (which depends on the current through that conductor). Constant velocity, thus, can only be achieved by the previously accelerated electrons losing energy every now and then (by them bumping into atoms in the resistor).
A resistor relates a constant current to a specific constant voltage (i.e. an electric force field, as mentioned above). But if there is no resistance, current increases continuously (like velocity does) for constant voltage.
As to your method of switching: yes, that is a method of reducing (average) current without increasing resistance. It is practically used for dimming LED's for example. But the point is: if you switch the circuit on, it is not without resistance (LED's for example have a differential resistance $dU/dI$ due to their semiconductivity). Resistance will just have a lower value than what you needed if you wanted to limit current by an additional resistor.
If resistance became negligible, other factors become important in constraining current, like capacitance and inductance. Especially inductance becomes more and more important if your switching frequency becomes higher and higher. For very high frequencies, a conductor even significantly acts like an antenna, radiating "into the universe" the information (and energy) about the electrons being accelerated.
Of course, you could also apply switching to a superconductor, for example, but then again, inductance and capacitance would remain as current limiting factors.
Even if you apply switching hypothetically to free electrons (no superconductivity necessary), you will increase disorder in the electron "cloud" by repeatedly accelerating and decelerating them, and this disorder is nothing else but heat. It will not be possible for you, to regain during deceleration all the energy you put into the electrons during acceleration (in the sense of a reversible thermodynamic process). The most important reason for this is that the electrons change their state of motion due to radiation, which you can't control because it is omnidirectional and has infinite degrees of freedom.
To make a long story short, if you switch the current on and off, you will eventually do nothing else than the atom bodies do in a resistor: dissipate energy of the electrons, in order to keep them at an average velocity instead of accelerating. Stop the charges more often and you dissipate more energy (higher resistance). Stop them less often and you disspiate less energy (lower resistance). Never stop them and you get free charges accelerating continuously.
A: This is possible for alternating current (AC). For AC voltage sources the definition of resistance is extended to what is called impedance. If you supply a voltage of the form
$$V(t)=V_0\sin(2\pi f t)$$
then for simple elements (resistor, inductor or capacitor) the current will be of the form
$$I(t)=I_0\sin(2\pi ft+\phi).$$
Here $f$ is the frequency and $\phi$ is some phase offset. To make the math easier the current and voltage are often treated as complex numbers but you don't need to understand complex numbers to understand impedance. The impedance is then defined as
$$|Z|=\frac{V_0}{I_0}.$$
For a resistor we get $|Z|=R$ and a phase offset of zero i.e. $\phi=0$. Capacitors and inductors also have impedance (in addition to a phase offset) but they do this without wasting energy. In reality they will always waste some energy but ideal capacitors and inductors have zero losses. The impedance of capacitors is $|Z_C|=\frac{1}{2\pi f C}$ and that of inductors is $|Z_L|=2\pi f L$. Because impedance and resistance have similar formulas these components basically act like resistors but without wasting energy. For example if you want to step down the voltage from your outlet you would rather want a capacitor/inductor since a resistor would waste a lot of energy.
A: The size of the resistance is not directly correlated to the size of the energy loss. With zero resistance, you have no excess heat loss - with infinite resistance you likewise have no heat loss. The heat loss is describe as the power $P$:
The reason is that a typical, so-called ohmic, resistor follows Ohm's law with the voltage $U$ typically being a restricted variable.
$$U=RI\qquad,\qquad P=IU$$

*

*For zero resistance (replacing it with a wire piece) there is no voltage drop $U$ across it. So, the power - the heat production, Joules per second - across it is $P=0$.


*The possible voltage $U$ will always be limited by the voltage source. So, for an infinite resistance $R\to \infty$ (corresponding to cutting the wire or replacing with an insulating material), since $U$ is fixed at its upper limit, the current $I$ must be zero $I\to 0$. Thus again no power, no energy loss.
The power-to-resistance graph of an ohmic resistor will thus look somewhat like a downwards parabola. For increasing resistance it starts at zero, grows to a soft top and dips gradually to zero again.
Only between these two zero-points do we have any power production. If you want a resistance value somewhere in that interval, you will always have heat loss if you keep using an ohmic resistor. This component slows down the flow of electric charges via "physical" blockade in a less-conductive-but-not-insulating material. Charges "bump" into atoms within the material structure and are thus delivering a bit of their kinetic energy to the material itself - which converts into thermal energy, a temperature increase, and thus heating of the material. To avoid this, you must use another type of charge-resisting method than a typical solid resistor.
A: If there is resistance then there is momentum being passed from electrons to other electrons and atomic nuclei which we call heat.   In superconductors electrons pass some momentum to positive charges but  also get it back because superconductors have charges that move in a non random coordinated manner.
