Can British society exist and survive without the monarchy? Can it do without the royalty?

Of course it can - but I doubt you'll get rid of them soon. The monarchy and the royals bring a lot of tourists, and tourist money, into Britain, even though they also cost a lot to maintain, they do a lot of charity work, and they act as effective symbols of Britain abroad. In addition, because of the whole historical, traditional aspect of the monarchy, many people would be reluctant to be rid of it.