1) Citizen / LPR :
LPR do not have the same rights. Being a LPR, you cannot vote for major elections. You're supposed to be able to vote for local elections.
As a LPR, you can work for almost any jobs (see later), have as many jobs you can. You can also leave your employer. You no longer need a sponsor.
Usually, jobs for the government are only for US citizens.
LPR status is a priviledge : you have to comply with US laws, or you could be deported.
I hope I did not forget to mention something else. But see USCIS website:
http://uscis.gov/graphics/citizenship/welcomeguide/index.htm
2) A LPR can become a Citizen: the process is known as Naturalization. ( see experiences under the appropriate section of this website)
After you become a citizen, you have the same rights as a " US born" Citizen except that you can not become President .
A general rule is that you can apply for citizenship 4years and 9 months after you were granted your GC ( this period varies if for example you go back to your home country with a reentry permit ( <2years))
3) First, if you go back to your home country, pay attention to the length of your stay ( so that you do not loose your GC, i.e. LPR status).
If you wish to maintain LPR status, you have to maintain
permanent residency in the US .
You say you'd like to go back to Nepal for one or two years : you would NEED a re-entry permit. But it does not even guarantee that you wuld be admitted back, depending on your ability to show ties to the US. ( see the posts about that for more detailed info)
Now, as a LPR if you marry someone not legally in the US, it takes approx. 5years for you to sponsor your spouse so that he/she can come to the US.
If you were a citizen, this period is 3 years ( but it takes 5 years to become a citizen...).
I you had married your girlfreind before getting your immigrant visa, she would have been able to get also a visa. ( see the posts on this, including my own experience : Paris, France ) But, now it is too late for this option.
Another possibility would be to try to get a nonimmigrant visa for her ( a student visa for example ).
However, this is
illegal : to get a nonimmigrant visa, you have to prove that you do not have immigrant intent ( which would not be the case).