Should I Impute target values?
For the missing data problem, one thing to be aware of, is the missingness mechanism. Depending of the dataset, the NA's (Missing Values) you have could be a result of a condition of the phenomenon and you shouldn't impute then using mean but maybe. 

Besides, for the dependent variable, if you want to train a model with the independent ones to predict this, let's say Y, you wouldn't train a model using this observation with NA on the dependent (target?). Then, you would drop this lines or maybe using another technique which takes into account the dependence of the other variables.

I think a good start is to give look at this:
Missing-data imputation

It shows the limitations of using some approaches like yours and defines the mechanisms of missing data.