As a Green Card Holder you are basically saying that you are living in the U.S. not is another country. If you stay outside the U.S. too long the government may believe that are no longer living in the U.S. If they determine that you are no longer living in the U.S. (Abandoned your Residence) they can have your appear in Immigration Court and take away your Green Card. If you know that you will be outside the U.S. for over 6 months or more you can ask for permission to be out longer in Advance. This still does not guarantee that you will not be stopped and questioned by U.S. Customs when coming back to the U.S. It really depends on your personal situation.
If you received a Green Card through Asylum and travel to the country that you asked for Asylum from you can also run the risk of ending up in Immigration Court to have your Green Card taken away. If for some reason you have to go back to the country you were granted Asylum from you should talk to an Immigration Attorney about your situation.
In regard to becoming a U.S. Citizen, after you have had your Green Card for the required period of time, your time outside the U.S. and the Countries you traveled to will come up again. People have applied for Citizenship and have instead received a Immigration Court Notice. You have to be in the U.S. generally half the required time period before you apply for U.S. Citizenship.
Please contact a licensed Immigration attorney before you act on your specific situation. Immigration Laws and Federal Regulations are constantly changing and the general information provided can change over time.