How do long rivers exist?
Well, I can only agree that it is indeed amazing and it doesn't get less amazing when some geology, hydrology and geomorphology is added to the the amazement. 

All precipitation that falls on land must flow back to the oceans somehow, and unless it evaporates it will flow in rivers or as groundwater towards a lower hydrostatic level. Water can not flow uphill, so it follows (and crafts) the slope in topography. Water will flow from mountains towards the ocean and due to geometric and morphological reasons, rivers joint to a drainage pattern due to the geology and topography.

Mountain ranges are often formed from orogeny, where tectonic plates collide. Rivers starts at high altitude, radiating out in all directions from mountains, but as collisions in the present continental settings are often on the rim of large plates (e.g Andes, Alps, Himalaya), and the rivers can't cross the range, it will have to travel the whole continental plate to reach ocean level. In the case of Asia, most large rivers starts in Himalaya (or other tectonic active regions, as Altai), in Europe large rivers starts in the Alps. In Africa, they start in the tectonic active rift zone. 

This map shows the ocean drainage dividers. The border between the drainage areas are the line where rivers start. 



Lakes could be understood as temporary stop in the flow, and it will quickly fill up with vegetation and sediments as the velocity of the water decreases. In a geologic context, lakes are never very old, Baikal being the oldest, 25 million years. Drainage patterns are often much older. 

So why are some rivers so long? Because of tectonic uplifts and orogenies. The answer to your question 'Did the rivers make the downhill paths themselves?' is yes, as erosion works toward the peneplain (low relief).