I need help with a two-part assignment. 1. Does Socrates teach anything of his own in the Euthyphro? What textual proof can you offer to support your point? 2. Explain and summarize the 2nd definition of piety and Socrates’s response to that definition.

The first question is itself somewhat problematic. If you read the Euthyphro, you will note that the method of the dialogue seems to be one of offering up a series of questions and tentative solutions. The tentative solutions are then examined and rejected. Thus while Socrates examines many different definitions of piety, he does not set forth any single definition as dogma. The dialogue ends with an aporia, as can be seen in the final statement "Then we must begin again and ask, What is piety?" Of the ideas examined, of course, many are original. One of the most interesting aspects of the dialogue is that it probably is the first example of a Greek text to use the term "ousia" to mean "essence". Although the nature of piety itself is not resolved in the dialogue, one can answer that the distinction between essence and "not-essence" (accident) is established and not refuted; thus Socrates could be said to teach philosophical method in his call for definitions to reflect the ousia of what is being defined. 
The second definition of piety is that it is what is pleasing to the gods. Socrates refutes this with the observation that different things please different gods. For example, chastity pleases Artemis but not Aphrodite. 
https://plato.stanford.edu/entries/plato-ethics-shorter/