I agree with my colleagues. You should consult an experienced immigration attorney for this matter. You may schedule a consultation with an immigration lawyer in your area, my firm is handling these matters in New York. If you would like free legal updates on these immigration issues you may sign up for our newsletter at http://www.shautsova.com .
You would need it only if your green card was based on employment. So, you don't need to bring all your I-20s. But you can, if you want. Main thing: you need to prove that you entered the country legally: with visa, and you were inspected.
To qualify for DACA you need:
Individuals who meet the following criteria can apply for deferred action for childhood arrivals:
■are under 31 years of age as of June 15, 2012;
■came to the U.S. while under the age of 16;
■have continuously resided in the U.S. from June 15, 2007 to the present. (For purposes of calculating this five year period, brief and innocent absences from the United States for humanitarian reasons will not be included);
■entered the U.S. without inspection before...
If the petitioner does not meet the financial requirements, he needs to find a joint sponsor who does. Keep in mind, that joint sponsor's family size is important. It's best if you have an attorney helping them.
You might be able to come back with a waiver. You need to contact an attorney, who will review documents about your criminal conviction and deprtation, and will be able to tell you what exactly you need to do.
IF one of your recent trips was for longer than 6 months, I advise you to meet with an attorney before applying. Search for somebody in your area or somebody you are comfortable with. AILA website or AVVO are good sources for good attorneys.