I) The gas law under investigation is Gay-Lussac's Law (also known as the Pressure Law).

ii) To ensure that all the air trapped in the bulb was at the temperature recorded by the thermometer, the thermometer should be placed inside the water bath, close to the bulb. Additionally, the water in the bath should be stirred continuously to ensure uniform temperature distribution throughout the water and the air bulb.

Given:
Height difference in mercury $h = 20 \text{ cm} = 0.20 \text{ m}$
Density of mercury $\rho = 13600 \text{ kg/m}^3$
Acceleration due to gravity $g = 9.8 \text{ m/s}^2$
Assume standard atmospheric pressure $P_{atm} = 1.013 \times 10^5 \text{ Pa}$.

The graph of pressure versus temperature (in $^\circ\text{C}$) for a fixed mass of gas at constant volume (Gay-Lussac's Law) is a straight line with a positive slope. It does not pass through the origin $(0,0)$ because $0^\circ\text{C}$ is not absolute zero. If extrapolated, the line would intersect the temperature axis at approximately $-273.15^\circ\text{C}$.

(Note: The graph should be a straight line with a positive slope, starting from a point in the positive temperature range and extending upwards. It should not pass through the origin.)