Yes. You may travel to Mexico with an approved extension and expired visa in your passport under the contiguous territory rule.
Mr. Asatrian's practice is dedicated to the area of immigration and nationality law. Please note the information provided herein does not constitute legal advice and should not be construed as such by anyone. It should not be relied upon as legal advice as more specific facts, research and analysis may be required to formulate a proper strategy and action in your matter. Please note this does not create an attorney-client relationship whatsoever. You should seek the assistance of an experienced immigration attorney to review your matter thoroughly and gather all of the necessary information and documentation to provide you with the best possible legal solution.
1. Mexico yes; "south America": you will need a new visa stamp to be readmitted.
2. See my answer 1. above. You can be readmitted on your I-797 approval notice when coming back from an adjacent country (Mexico, Canada) on a trip of less than 30 days, for anything beyond those two countries you'll need to apply for and obtain a consular visa "stamp" to be readmitted.
3. Carry copies of everything to produce if asked either at the port of entry or airport upon return.
Kindly be advised that the answer above is only general in nature cannot be construed as legal advice, given that not enough facts are known. It is your responsibility to retain a lawyer to analyze the facts specific to your particular situation in order to give you specific advice. Specific answers will require cognizance of all pertinent facts about your case. Any answers offered on Avvo are of a general nature only, and are not meant to create an attorney-client relationship.
To travel to Mexico or Canada, you can travel for up to 30 days without a visa. To travel anywhere else about, you will need an H-1B visa in order to reenter the U.S.
Mr. Shusterman is a former INS Trial Attorney (1976-82) with over 35 years of immigration experience. His response to your question is general in nature, as not all the facts are known to him. You should retain an attorney experienced in immigration law to review all the facts in your case in order to receive advice specific to your case. Mr. Shusterman's statement above does not create an attorney/client relationship.