Is hour of day a categorical variable? Is "hour of the day" where the value can be 0, 1, 2, ..., 23 a categorical variable?  I would be tempted to say no, since 5, for example, is 'closer' to 4 or 6 than it is to 3 or 7.
On the other hand, there is the discontinuity between 23 and 0.
So is it generally considered categorical or not?  Note that 'hour' is one of the independent variables, not the variable I'm trying to predict.
 A: Depending on what you want to model, hours (and many other attributes like seasons) are actually ordinal cyclic variables. In case of seasons you can consider them to be more or less categorical, and in case of hours you can model them as continuous as well. 
However, using hours in your model in a form that does not take care of cyclicity for you will not be fruitful. Instead try to come up with some kind of transformation. Using hours you could use a trigonometric approach by
xhr = sin(2*pi*hr/24)
yhr = cos(2*pi*hr/24)

Thus you would instead use xhr and yhr for modelling. See this post for example: Use of circular predictors in linear regression.
A: Hour of the day isn't best represented as a categorical variable, because there is a natural ordering of the values. Hair color, for example, is categorical, because the ordering of the categories has no meaning - {red, brown, blonde} is as valid as {blonde, brown, red}. Hour of the day, on the other hand, has a natural ordering - 9am is closer to 10am or 8am than it is to 6pm. It is best thought of as a discrete ordinal variable. It has an added characteristic of being cyclic, since 12am follows 11pm and precedes 1am.
A: Theoretically, it depends on how you format the variable i.e. it can be "continuous" (modeled with a single coefficient) or categorical (a coefficient per "hour" of day). You could also do a mix of both e.g. piece-wise functions.  
Practically, because 0 and 23 is essentially the same "hour" of day, I would consider grouping periods of the day into larger, more homogenous and credible groupings. For example, in 8 hour increments - 8am-4pm, 4pm-12am, and 12-8am. 
