Can you not just look at the incoming emails into
email@example.com? If you use the Gmail search bar to search for
to:firstname.lastname@example.org you will see all incoming email to the dummy address.
The other thing Gmail will do is automatically add contacts to your contacts list for any emails to which you reply, reply-all, or forward. Here's a link to an article on automatically adding contacts: http://support.google.com/mail/bin/answer.py?hl=en&answer=8933 (Check out the 'Automatically added contacts' section)
For each of these contacts, if you reply with a message like "Email receipt confirmed," then it will automatically add the email address into your contacts. They will appear in the "Other contacts" section of the left-hand navigation.
An even more automated way to reply is to create a filter that will look for emails coming into the inbox and auto-respond with a "Canned Response" More about doing that here. If you were set up this filter to send a canned response, presumably all of the appropriate messages will be replied to with a Canned Response like "Email has been added to contacts" (Canned responses are written by you)
Lastly. if you need a quick and easy way of recording all the students email addresses without using Gmail to capture it all, you can set up a Google Form. This will allow you to set up a quick webform that asks users to enter their preferred email address. All answers will be stored in a Google Spreadsheet that you can then use to summarize the data.