|MadSci Network: Earth Sciences|
You have about three questions in there, so let's address them one by one (not quite in the order you asked)
Is the Earth perfectly spherical?
No, the Earth has a bulge at the equator exactly equal to that it would have if it were liquidbulging out due to centrifugal force ( I know that centrifugal force doesn't really exist from a strict scientific sense. It's only an inertial effect as we move in a curved path, However, we can feel it in a fairground ride and it makes many things easy to understand )
The effect of this is that the Ocean at the equator is 20km further from the centre of the Earth than at the poles. The Earth's "solid" surface is also bulged out, otherwise all the equatorial continents would be completely flooded by this ocean. The reason for this is not that the solid Earth was once softer, but that geological time is so long that even "solid" rock can bend and flow if the force remains for millions of years.
That was another of your questions - is the Earth solid?
The answer - Yes, AND No!
The shape of the Earth is, to first approximation a slightly flattened sphere. It's surface is an "equipotential surface" - things on that surface have the same energy so water cannot flow sideways to a lower point. Rivers do not creep towards the equator by centrifugal force because the rocks and the oceans already did so millenia ago and made a hill that exactly balances the centrifugal force.
Rivers flow downhill. Whichever way is down, there they go. The Amazon is a great example that beats your theory. It is near the equator and flows from the Andes in the west to the Atlantic in the east. Locally, rivers can get trapped in mountain belts and have to flow parallel to them before they can get round the end. Look at the Ganges and the Brahmaputra that have to flow around the Himalayas. The upper reaches of the Amazon flow in a pattern of North-south sections between mountain ranges.
Look at the continents where your rivers are. See the mountains and the plains, reaching down to the sea. The rivers flow downhill. In North America, there is a mountain range in the west (the Rockies) and one in the east (the Appalachians). The centre of the continent is pretty flat. The Mississippi flows south down this great (nearly) flat plain until it reaches the Gulf of Mexico. It can't flow uphill and get out to the Pacific or the Atlantic. In the north - Canada - other rivers flow out north and eastwards to hudson's bay and the Atlantic, although the great lakes confuse drainage patterns and there are a lot of funny things up north related to glacial rebound - but that's another question...
Try the links in the MadSci Library for more information on Earth Sciences.