The I-V characteristics of materials and devices should always be measured at the same thermodynamic conditions, i.e. at the same temperature. Mixing the actual isothermal I-V characteristic with the temperature dependence doesn't lead to any useful data for the purposes of physics (but it is occasionally done in electrical engineering and electronics design for certain parts like NTC heaters and breakers).
A pure semiconductor at a constant temperature would be a pretty good Ohmic conductor, i.e. the current will be proportional to the applied voltage. This is a lot harder to measure properly on semiconductors than on metals, though, because of junctions formed with the metal wires that one has to attach for the measurement.
The conduction characteristics of semiconductor devices with one or multiple different materials forming junctions, on the other hand, is highly non-linear and can be made very complex. These devices will also have a temperature dependence, but it can be tuned very finely with appropriate material combinations and geometries.
Pure metals have typically increasing resistance with increasing temperature, but alloys can be made that have almost constant temperature characteristic (i.e. they are both Ohmic and temperature independent). One can also make metal alloys with negative characteristics, if necessary. Both constant and negative temperature characteristic is of enormous importance for the design of electronics, almost none of which would function properly if we couldn't make these near zero-TC metal alloys for resistors and NTC's for temperature measurement and compensation.
Non-metallic materials with very strong negative temperature characteristics often use percolation phenomena, i.e. on grain boundaries in sintered crystal powders, where conduction can only happen in very few narrow points in the material. As the material expands, these points of contact may get lost and the resistance may increase by many orders of magnitude over the technical temperature range of the material. The physics of these systems is very different from that of metals and semiconductors.
I think it would be better to say that power lines are designed to avoid ohmic heating rather than that they make use of it. I am not sure about the potential advantages of the heating for lines that may otherwise be weighed down and damaged or destroyed by snow and ice in cold climates, though. One would have to look at the design requirements for these power systems to understand if their designers make explicit use of these otherwise unwanted losses.
You are correct that one can trade current for voltage and vice versa by adjusting the resistance in circuits. Much of electronics design is a repeated application of that principle.
As for the question of how to design materials that have nearly temperature independent characteristics, that would require a very deep dive into solid state physics and materials research and I will leave that to someone who actually has the necessary detail knowledge. The guiding principle in many of these practical applications is that one tries to offset a positive gradient of one material with the negative gradient of another or one tries to combine multiple materials in such a way that the physical effects (like the formation of defects in the mixed material) offset bulk effects like the increase in the number of conduction band electrons in either of the constituents.
Best Answer
For a given current, the heat loss in the transmission line is indeed proportional to the transmission line's resistance:
$$P_T=I^2 R_T\ \,$$
where $P_T$ is the power lost in the transmission line as heat, $I$ is the current, and $R_T$ is the resistance of the transmission line.
The transmission line's resistance actually isn't very high; the question arrives at that incorrect conclusion by using the wrong voltage in Ohm's law. What would be valid is
$$R_T=\frac{V_T}{I}\ \ ,$$
where $R_T$ is the resistance of the transmission line, $I$ is the current, and crucially, $V_T$ is the voltage across the transmission line, not the 400 kV source voltage. The 400 kV source voltage is the sum of $V_T$ and the voltage across the load,
$$V_S=V_T + V_L\ \ ,$$
and $V_T \ll V_S$, i.e. $V_L \approx V_S$.
According to Watt's law, the current through the load is
$$I=\frac{P_L}{V_L}\ \ ,$$
where $P_L$ is the power needed by the load. (For simplicity, I'm treating the load as being purely resistive, with no capacitive or inductive reactance.) But the current through the load is the same as the current through the transmission line, so you can plug $I$ into the first equation above to give
$$P_T=\left (\frac{P_L}{V_L}\right )^2 R_T\ \ .$$
I.e., for a given $P_L$ and $R_T$, the way to make $P_T$ be small is to make $V_L$ be large, which means you need to make $V_S$ be large.