Impute phenotype under some constraints
Question 1: Is there a tool for imputing phenotypes while also using genetic data?

Yes, try Andy Dahl's Phenix from the Marchini lab.  

Disclosure: I did my PhD in the Marchini lab, although I was not directly involved in this research.

Question 2: Is there a way to impute categorical data using methods intended for continuous data?

If the variable of interest has only two categories, then you should be able to simply recode the categories as arbitrary (differing) numbers and get sensible results. But do make sure you understand the tool you are using to see if this will work.

For more than two categories, if your imputation approach allows the simultaneous imputation of multiple variables, you could try recoding your categories as contrasts or one-hot.  After recoding, you could impute the variables resulting from recoding together.