I would like to know if it makes sense to run an ordered probit regression (dependent variable is ordinal with three outcomes) with only categorical explanatory variables (some are dichotomous e.g. gender; others are categories that have been entered as dummy variables e.g age).
(Partially answered in comments, summarized below)
Like in about any kind of regression model, you can use any kind of variables as predictors. Categorical variables cannot be used directly, they must be encoded, but you can do that in exactly the same way you use for linear regression. So just go ahead!