No, they must have a median. However, if the data set is of even order, the median may not belong to the data set.
For example, the median of 1,2,3,10 is halfway between 2 and 3 or 2.5 which is not a data point.
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To report what color something is would be qualitative data not quantitative. Quantitative data, by definition, is a measurement of some quantity, and therefore it requires a number of some kind. Light intensity can be measured with a photometer, which would be quantitative data. Color isn't.
What are some distinct advantages of a qualitative data gathering strategy, such as participant observation, over more quantitative approaches
Organizing the data into a frequency distribution can make patterns within the data more evident.
Organizing the data into a frequency distribution may make patterns within the data more evident.
Yes, but only if the data set includes some negative numbers. For example, recording winter temperatures at the South Pole. They will all be negative (or most of them will be) so the mode and the median will be negative as well.