Hello,

If someone who is not a U.S citizen and do not hold a valid passport
but are a permanent citizen of the united states and are recognized by
the U.S. that he/she is a permanent citizen of the states. This person
also has a green card. Can this person?s go outside of the country for
a vacation and would not be bother by the government for entering and
leaving the country? If so, what requirement or "stuff" do this person
needs to get in order to legally leave the country for a vacation and
still be able to go back into the country without any hassle.

Also if you are a legal citizen of the U.S because you were sponsored
by the U.S. to come to stay in the country but wants to leave for a
vacation but do not have a U.S. passport or valid green card. What can
this person do to be able to leave the country legally so that they
can come back into the country without any hassle?

Please provide in detail as to how one can get all of these legal
papers and provide the steps necessary to achieve this goal.

Thank you

Your terminology is wrong.  You mean to say "permanent resident" or
"resident alien".  To return to the U.S., this person needs a passport
from their country of citizenship, along with their "green card".  If
the person is on a visa, then they need that visa to return (in
addition to the passport, but in place of the green card).  This
person should be careful not to run afoul of any requirements to
remain in the country for a given period to maintain their status.