Changes in volume
The chemical reaction for burning a candle is something like this:
$$\ce{C25H52 + 38 O2(g) -> 25 CO2(g) + 26 H2O(g)}$$
For every 38 dioxygen molecules used, you are making 25 carbon dioxide molecules and 26 water molecules (which start out as a gas, but will condense once they reach an area of lower temperature such as the glass surface). So once the flame is out and the water has condensed, the volume should be less than that of the fresh air (20 % of the volume initially are oxygen. Its volume will be replaced by that of carbon dioxide, so the volume will go from 100% to 25/38 * 20% + 80% = 93%). While the candle is burning, however, the temperature is higher, and some of the water will be in the gas phase. Also, while the air was already hotter than room temperature before the container was covered, the temperature will rise as soon as the influx of cooler (fresh) air is cut off.
How hot will it get?
A tea light produces heat at about 30 J/s, and (with a molar heat of combustion of about 15,000 kJ / mol) used up $\pu{80 \mu mol}$ oxygen per second. This means you need about 10 mL of fresh air each second the candle burns. The container has a volume of about 400 mL, so the candle should burn <40 seconds. In the video, it burned about 2 minutes (not bad for an estimate). Using up all the oxygen in the container would give off about 1.2 kJ, which is sufficient to raise the temperature by 1000 degrees Celsius if it were isolated. As it is, most of the heat flows into the container.
What is different when using water?
Water has a higher heat capacity, so it is more efficient in cooling down the gas (and the container). As a consequence, the water level will rise appreciably. As MaxW mentions in the comments, the plastic probably did not seal perfectly. The pulsing could be intermittent flow of gas, or temperature fluctuations because of mixing.