This post is split into two parts. This post is completely spoilers so I didn't bother to put under spoiler tab. But, I tried to give a spoiler-free title because I don't want someone getting spoiled. The post discuss the events in the 2nd season of Higurashi. Here the Oyashiro-sama's "curse" refers to cycle of life and death experienced by the Hinamizawa residents.
Part II of post: How and when was Rika Furude born?
In the 2nd season, we get to know about Hanyuu who is the true incarnation of Oyashiro-sama. She has the ability to manipulate space-time to create alternate dimensions, parallel to the original Hinamizawa. This power is the cause for the eternally repeating June of 1983. Every time Rika dies, Hanyū uses her ability to escape whatever world they are presently in. The two then arrive in an in-between dimension. Eventually, a new Hinamizawa is generated where Hanyuu and Rika enters into it and a new chance is created to prevent Rika from dying.
If Hanyuu can foresee everything and can create worlds in order to create a chance for Rika to prevent her death, why can't she just create a world where Rika never existed? In this case, this cycle would never had occurred and Hinamizawa residents wouldn't had faced the consequences. Why is Rika so important to Hanyuu?