Missing outputs in multiple-output neural net
Since we are talking about multiple different types of targets (classes versus numerical for example) we already need a composite loss function. I will consider how to balance the different composite parts of the loss function outside of the scope of this answer but if you look into multi-task learning there are solutions to this. What you could do (both in training and evaluation) is to mask the parts of the loss function that are unknown. 

You can do this fairly easily (but hacky) by passing weights as input during training or prediction for each output in your sample whether it was available or not as 1 or 0 and multiply that component with the weight. For analysis of the training and prediction you need to make sure to take the mean of only the available labels because otherwise your loss values are too optimistic.

I do think this approach is very valuable in a lot of cases, this can also help with semi-supervised learning where you do have labels available for a similar task.