Why do only thick wires have less resistance

The wire gauge you need is a function of several things.

• The acceptable voltage drop or power loss (that appears to be the only thing considered in the website you linked). The voltage drop (and power loss) is proportional to wire length and inversely proportional to the cross-sectional area of the wire- in other words inversely proportional to the square of the wire diameter (assuming constant current).
• The acceptable temperature rise. This is a function of the number of current-carrying wires bundled together, the environment (maximum ambient temperature and air pressure or altitude, for example), the insulation type, the wire type (some types of wire are plated to withstand higher temperatures than bare copper without corroding).
• Regulatory requirements and other considerations- for example, the wire may be rated for 200°C insulation, but you might not want the wire to run that hot.
• Fusing- the fuse or circuit breaker should protect the wire in the case of faults such as overload or short circuit.

Very short lengths of wire can depend on heat sinking through the ends (indeed, in a vacuum, that may be the main heat loss mechanism), but usually that's not taken into account.

Normally you'd run through a checklist such as the above to make sure ALL the requirements are satisfied simultaneously, so you might find that using PTFE insulated wire allows you to use AWG 18 wire, but because of the voltage drop limitation you'll have to use AWG 12 wire.

It's the energy-band model. The idea is materials have separated conduction and valence bands in which electrons may reside.

Eg is the size of the gap between these bands. In metals, the gap is zero, the bands overlap. In insulators, the gap is so huge that it is impossible for electrons to reach the valence band (temperature >>1000K needed). Semiconductors are inbetween, the gap is a few eV (Si: 1eV@300K).

In think this "intrinsic region" should just mean "temperature-depending conduction" in contrast to the conduction introduced by doping atoms. The latter change the energy niveaus of the bands and the size of the band gap. When the temperature is sufficiently low, there are other effects which govern the resistance measured instead (meaning: it's still there but you cannot measure it independently.)