Assalamualaikum I want to know is it possible that I can stay at my place and she can stay at her place right after the actual nikah (After signing the marriage contract) until the time comes? Cause I don't want to have a Haram relationship with her. And I am in love with her. So I want to have a halal relationship. But we are still really young and still am a student (I am still just 18) so I was wondering if it was possible that I can make the relationship halal but stay at my place and she can stay at her place and when the time is right we can arrange a traditional marriage ceremony or feast few years after the actual nikah. Is that possible?
Yes it is permissible in Islam. As there's no shari'a ruling prohibitting it.
Note that when the Prophet () married 'Aisha she lived in her parents home several years before moving to her husbands home (thanks ti @Ahmed for pointing at this in the comment):
The Prophet (ﷺ) engaged me when I was a girl of six (years). We went to Medina and stayed at the home of Bani-al-Harith bin Khazraj. Then I got ill and my hair fell down. Later on my hair grew (again) and my mother, Um Ruman, came to me while I was playing in a swing with some of my girl friends. She called me, and I went to her, not knowing what she wanted to do to me. She caught me by the hand and made me stand at the door of the house. I was breathless then, and when my breathing became Allright, she took some water and rubbed my face and head with it. Then she took me into the house. There in the house I saw some Ansari women who said, "Best wishes and Allah's Blessing and a good luck." Then she entrusted me to them and they prepared me (for the marriage). Unexpectedly Allah's Apostle came to me in the forenoon and my mother handed me over to him, and at that time I was a girl of nine years of age. (Sahih al-Bukhari and Sunan ibn Majah)
This basically supports the permissibility of doing so. But usually the husband must be in charge of his wife, which seems unlcear in your case. So as you are not able to be in charge for your wife this marriage (with such a long duration of stay in the parental home after the 'aqd an-Nikah as described in your question) has a very weak basis in some madhhabs this is a reason for invalidating a marriage. So it strongly depends on whether the guardian of your to be wife accepts these conditions, your to be wife accepts them and your family accpety to support this.
Note that the wife has to obey her husband only after leaving her parents home and moving to her husband's home and there in only after consummation of the marriage.
I recall that sheikh az-Zindani propsed this for Muslims leaving in non-Muslim countries as an option to help them avoid zina, as many young people there live in unlawful relationships he or the author of the article I read his fatwa in even called it "zawaj friends" (marriage-friends).