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It looks like a Polish name with a German spelling. I can tell you that Rumschik hardly exists in Germany, and Rumsik (Polish spelling) doesn't seem to exist in Poland. Whatever your origins your name is a rare one, which usually means it's been overlooked by the experts.
There is a Polish name, Rumszek, not common. There is also Rumsicki, pronounced RoomSHEETskee, which could denote someone from a place called Rumsik, though no such place has been identified.
There is a place called Romischek on the Austrian-Slovenian border, but as the surname doesn't appear to exist in Austria that seems an unlikely source for it.