I have completed the graduation requirements for the MS degree. But I would like to stay back in the university for another semester to gain some research or project experience so that I can get a very good job. Is this possible?
You are required to graduate as soon as you complete the requirements for the degree. You can always take additional classes as a non-matriculated student after your graduation or even informally work with a professor assuming you have a working relationship with a professor already. Note that if you are an international student, you may require appropriate immigration authorization for this. But in no case you can delay your graduation.
I am an international student and this is my final semester in the MS program and I need only a few credits to graduate. Do I still need to register for 9 credits as a full-time student?
You can take an underload in your last (graduating) semester. This means registering for less than required number of credits and just enough needed for graduation. You will still be considered full time. However, this requires prior approval. See the graduate handbook. However, you should really make sure that you indeed graduate in that semester. There can be serious immigration related implications if you do not for some reason (e.g., you fail a course). If you are not confident, it is best to register the required 9 credits.