시민권

I am better at typing English so I am writing to you in English.  I am fluent in Korean, both in reading and writing, so please feel free to reply back in Korean - that way my mom can read it too...

This is for my mom.  She is 74 years old and has been a green card holder since 2013.  We are thinking about applying for the citizenship just so there is no limitation on how long she can be out of the U.S.  She will be traveling to Korea and it is very possible that she may stay in Korea for a long period of time.  

I have the following questions: 

1.  As a permanent resident, can she lose the permanent residency status by staying out of the U.S. too long?  Or staying out of the U.S. too frequently too long.  

2.  Is obtaining the citizenship the right solution for this? 

3.  Due to her age, I believe she is allowed to hold dual citizenship.  If correct, does she have to give up Korean citizenship when she obtains U.S. citizenship then apply for Korean citizenship again?  

I appreciate any advice/info you can provide.  Thank you. 

Jin