Halacha forbids cutting tissue paper on Shabbat. If a person did not prepare cut paper before Shabbat and finds himself in need of tissue paper, he may cut paper with a "Shinui," meaning, in an abnormal way, and not along the perforated line.
One is also permitted in such a case to ask a gentile to cut tissue paper for him on Shabbat, provided that he specifies that the paper should not be cut along the perforated line. Cutting not along the perforated line is forbidden only "Mi'de'rabbanan" – by force of Rabbinic enactment, and Halacha allows asking a gentile on Shabbat to perform an act forbidden "Mi'de'rabbanan" in cases of great necessity. (This Halachic principle is called "Shebut De'shbut Be'makom Sorech Gadol.") This situation, which involves basic hygiene and human dignity, would certainly qualify as a dire necessity and thus one may, when necessary, ask a gentile on Shabbat to cut tissue paper not along the perforated line. This is the ruling of Hacham Ovadia Yosef, as recorded in Yalkut Yosef (307:49), and of Rav Shemuel Pinhasi, in his work Ve'daber Dabar (p. 199).
If the lights go out in one's home, and a child in the home is frightened by the dark and cannot be calmed, one may indirectly ask a gentile to turn on a light. Rav Pinhasi cites this ruling from Rav Yaakov Haim Sofer (1870-1939) in Kaf Ha'haim (276:14). Although one may not explicitly ask a gentile to restore the light in such a case, he may indirectly express such a request (such as by saying, "The house is too dark") in order to calm the frightened child.
Summary: One who does not have cut tissue paper on Shabbat may ask a gentile to cut paper for him, but he must specify that the paper should not be cut along the perforation. If the lights go out and a child is inconsolably frightened by the dark, one may hint to a gentile that he wishes for the light to be restored.