I guess you have some working hypothesis. I would work like in the "good old time" when you had no cheap computing power and powerful stats programs at hand: regroup categorical variables that fit together on substantial grounds linked to your working hypothesis, not because they correlate.
In any case with n=13 you can run an exploratory analysis, but not something more complicated.
Once you have somewhat reduced the number of variables and gotten more meaningful "factors" you can work as in qualitative data analysis: take a spreadsheet and you sort the cases in the rows on some major, substantial issue and put your regrouped categorical variables in the columns and sort them, too, on the basis of your knowledge of the research field. Do you see a pattern ? Maybe you have to eliminate further variables. Are your conceptions now confirmed ?
If you work in political science (for instance) the cases might come from two different sub-samples: from countries that are democracies and others that are run by dictators. So, you will do a sub-sorting for democracies and another one for dictatorships.
You can find more of these ideas in Miles & Huberman, 1994, Qalitative Data Analysis. Sage .
A statistical program that might help you because it is very visual and you have so few cases is the free ViSta from Forest Young. See http://www.uv.es/visualstats/Book/
I am also somewhat puzzeld about the people that demanded you to work with this data set. Do they know their stats ?
If you explain the type of data set you work with maybe you get better answers.
I wish you good luck, and data with an easy structure and no missings!
When can you do this ... is whenever you like and some researchers are fond to trying to model complicated relationships of responses with continuous predictors by splitting the predictor ranges into bins or intervals and thus converting them into categorical predictors. Discussion on this site alone is usually insistent on the difficulties and even dangers of this approach. See e.g. What is the benefit of breaking up a continuous predictor variable? -- in which the title of the question does not indicate all the answers.
When should you do this ... is the larger question and one answer is thus reluctantly and as rarely as possible.
Rounding (in your case to multiples of 5 when the range is from 0 to 1000) as such is usually only a little worrying. In many fields some rounding when reporting continuous variables is conventional, at least historically, and often grows out of general scientific awareness that high resolution of reporting (more decimal places, or more significant figures) is spurious or only trivially informative. Thus in many fields adult age rounded in years, or people's heights rounded in cm or inches, or temperatures rounded to $0.1^\circ$ C are standard even when more precision is possible, and such rounding has not stood in the way of much good statistically-based science. The important detail is whether several distinct levels are discernible in the data. Thus it is natural scientifically and statistically to expect age to be measured in months when looking at growth of children, but in days or even hours when monitoring small babies.
Strong bimodality to the extent that you have almost what Edgeworth and Yule called U-shaped distributions is worrying, but the question remains what you do when you have it. Reducing a U-shaped distributions to extreme categories would however be throwing away information on the grounds that you don't have enough, rather like a poor person giving away all their money on the grounds that they have so little any way (the moral or religious grounds for the latter being outside the scope of this forum).
So the key is that just because a continuous variable is well represented only in terms of a few levels doesn't oblige you to treat it as categorical.
This question is often mixed up with a different one, whether a continuous predictor be entered into a model as it is measured, or via a transformation or via a representation in simple polynomials, splines, orthogonal polynomials or fractional polynomials. What can bite is that with just two extremes well represented it may be quite unclear how that variable's contribution is best measured. As usual, a modeller falls back on linearity in the absence of other information, but sometimes there may be compelling physical (biological, economic, whatever) grounds for something different.
At the extreme this might drive some researchers to a categorical representation, but it seems rare that there is no information on what should happen in the middle of the range. Thus I have encountered British hydrological data where drainage basins (catchments, watersheds) of two different orders of area (in square kilometres) were encountered, very small basins instrumented intensively by university researchers and much larger basins instrumented by the national organisations who manage flooding and water generally. Here the absence of intermediate areas is just a side-effect of how the data were assembled, and there is prior scientific knowledge from many studies that area should almost always be used via its logarithm (that is, it makes no sense to categorise areas, say as small, medium or large).
Skewness is key in this example and complicates matters here and elsewhere, but I think its effects are extra rather than in contradiction to the above.
Best Answer
There is, as far as I know, no taxonomy of variables that captures all the contrasts that might be important for some theoretical or practical purpose, even for statistics alone. If such a taxonomy existed, it would probably be too complicated to be widely acceptable.
It is best to focus on examples rather than give numerous definitions. Number of days is a counted variable. It qualifies as discrete rather than continuous, and it is possible that the discreteness is important, particularly if most values are small. Some statistical people might want to insist that only models that apply to discrete variables should be used for such a variable.
At the same time, it is often the case that models and methods treat such a variable as approximately continuous. Population size is a yet more obvious example. Human populations can be in billions and many procedures effectively treat such variables as continuous, regardless of the familiar fact that people are individuals.
In contrast, a variable such as temperature is in principle continuous, but as a matter of convention temperatures may only be reported to the nearest degree or tenth of a degree, so the number of possible values may be rather small in practice. This does not usually worry anyone; it would certainly be perverse to call such a variable categorical. There are some contexts in which the discreteness of reported temperature is important: in reading mercury thermometers by eye and guessing at the last digit, people show idiosyncratic preferences for or against certain digits of the ten possibilities 0 to 9.
Also, what do we do with categories? Answer: we count them. We count males, females; unemployed, employed, retired, students; whatever. So, often we are modelling category counts.
In short, discrete counts are a common kind of variable, as well as continuous and categorical variables.