Fill missing values AND normalise
I don't understand why you would like to fill values with zeros ! This would basically mean, "this guy, who is 170 cm tall, weights 0 kg" and would fool your network. In my opinion, you have two options:


discard missing values (the entire row): you end up with less but more consistent training data
if you really need these rows, then fill missing values with some heuristics: for example, give them the mean of their column, or apply a simple linear regression. Beware that this will add a bias in the learning process, but it would be definitely better than giving random values. 


At very least, if you have a lot of missing values, then maybe you should think of selecting a specialized model for partial training data. You actually don't tell us what network you are using, but you might modify it to handle missing values more intelligently than by filling missing values. 

Finally, the need of normalizing once again depend on your model (which you don't describe). But this would definitely come after the processing part.