How can you predict a combination of categorical and continuous variables with Scikit learn?
I don't think there's a builtin way.  There are ClassifierChain and RegressorChain that allow you to use earlier predictions as features in later predictions, but as the names indicate they assume either classification or regression.  Two options come to mind:

Manually patch those together for what you want to do. For example, use a ClassifierChain to predict each of your categorical targets using just the independent features, then add those predictions to the dataset before training a RegressorChain with the numeric targets.

Use those classes as a base for defining a custom estimator.  In that case you'll probably look mostly at their common parent class _BaseChain.  Unfortunately that also uses a single estimator attribute, whereas you'd need (at least) two, one classifier and one regressor.
I have a dataset with a large number of predictive variables and I want to use them to predict a number of output variables. However, some of the things I want to predict are categorical, and others are continuous; the things I want to predict are not independent. Is it possible with scikit-learn to, for example, mix a classifier and a regressor so that I can predict and disentangle these variables? (I'm currently looking at gradient boosting classifiers/regressors, but there may be better options.)