Do rivers and streams run south to north?

The equator has nothing to do with which direction rivers flow in. Water flows downhill regardless of where on the Earth you are. Rivers can flow from south to north, north to south, east to west, west to east etc. as long as its downhill

The direction in which a river or a stream runs, is a matter of the terrain, going from high places to sea level. It has nothing to do with the poles or the earth's rotation.

For example, the Daly river, in north Australia, and the Betsiboka river in Madagascar run from south to north. But I must agree that there aren't too many in the southern hemisphere going that way.