Why does dough and its ingredients need to be cold?
It's not drastically important in making it workable, it's more important in the texture of the finished result.

When you roll out pastry dough, you are created interleaving layers of fat and the flour/water mix. When you cook it, the fat melts, leaving pockets in the dough, causing it to form flaky layers. This results in a crisp, light pastry.

For this to happen, the fat must remain as solid as possible until cooking. If the fat melts and blends with the flour and water before cooking, you will have a dense, cardboard-like pastry. Also, warm pastry dough tends to become sticky and hard to work with.