There are two important facts to recall that will help answer this question:
1. The resistance of a segment of conducting wire is given by this equation:
R = ρL/A
ρ is the resistivity of the material making up the wire. This value is a constant that depends on the properties of the material. Resistivities for various materials can be found with a quick Google search.
L is the length of the wire.
A is the cross-sectional area of the wire.
From this equation you can tell that a wire's resistance will increase if it is made longer and/or thinner, and the resistance will decrease if it is made shorter and/or thicker. Mathematically speaking, the resistance is directly proportional to the length and inversely proportional to the cross-sectional area.
2. The other fact is that a conductor's resistance is also dependent on its temperature. Generally, as a conductor gets hotter, its resistance increases.
Let us now tackle the list of statements:
1. A shorter wire will allow electricity to move through at a higher rate than a longer wire.
According to the equation for a conductor's resistance, a shorter wire will have a smaller resistance.
Now recall that current is the movement of electric charges and Ohm's law:
V = IR
V is the applied potential difference between the ends of the wire.
I is the current.
R is the resistance.
Assuming you keep the potential difference constant, when you have a smaller resistance, you will have a larger current.
Statement 1 is correct.
2. A short, thick, cold wire is the best conductor.
According to the equation for a conductor's resistance, a shorter, thicker wire will have lesser resistance. A cold temperature will also help to keep the resistance low. A low resistance means a higher current.
Statement 2 is correct.
3. How well a material conducts current is an internal factor affecting resistance.
Statement 3 is correct, assuming the physical property in question is the material's resistivity. The resistivity is one of the factors in the equation for a conductor's resistance.
4. If you double the length of a wire, you cut the resistance in half.
According to the equation for a conductor's resistance, increasing the length of a wire increases the resistance. Statement 4 is false.
5. If you double the thickness of a wire, you cut the resistance in half.
According to the equation for a conductor's resistance, increasing the thickness of a wire decreases its resistance. Statement 5 is true.
6. Superconductors have no measurable resistance.
A superconductor by definition is able to conductor electric current with virtually no resistance. Statement 6 is true.
7. The higher the temperature of the conductor, the lower the resistance.
A conductor's resistance generally increases with temperature. Statement 7 is false.
8. The resistance in a wire with less thickness is less.
According to the equation for a conductor's resistance, making a wire thinner will increase its resistance. Statement 8 is false.
9. Thickness, length, and temperature are internal factors that affect resistance.
Thickness (cross-sectional area) and length are both factors in the equation determining a conductor's resistance. Temperature is also known as a factor that affects resistance. Statement 9 is true.
10. When a light is first switched on, the light bulb's filament has a lower resistance than after it gives off light for awhile.
A device that draws a current will generally heat up given sufficient time. This increases the device's resistance. Statement 10 is correct.