If, for example, I have the following requirements: 1. Dog is an Animal 2. Zoo has Animal(s) Do I still need the 2nd diamond connector (the lower one) as shown here:

The 2nd diamond connector is redundant. Here is some advice about UML class diagrams style: http://www.agilemodeling.com/style/classDiagram.htm
The 2nd diamond connector suggests that in addition to animals the Zoo also has specifically Dogs.
Hi before tackling UML use your common sense :)
Do you want to model that zoo have animals of different species ? Then you should not only put dogs but also cats and all other animals class that can exist in THIS zoo type (class) and just use one diamond connector.
- Don't use white diamond connector. Use either "composition" or "association". In your case, use "association". The author of "UML distilled" has following comments. 
- The second aggregation is not necessary. However, you can change your model in this way: - (1) Set the first aggregation's target role (at Animal side) "animals". - (2) Delete the second aggregation. - (3) Create another class DogZoo, generalize from Zoo. - (4) Draw an aggregation from DogZoo to Dog, with the target role "dogs {redefines animals}" 
Hope that helps.
 
         
                                         
                                         
                                         
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