Fascinating question! The answer to all your questions is yes! But let's break them down a little.

A river is a continuous flow of water at Earth's surface, driven by gravity. A small creek flowing down the mountains over Santa Barbara and the Amazon River are both rivers. Rivers help sculpt and shape Earth's surface by eroding, transporting, and depositing sediment (soil, rocks, and other debris). Because rivers flow downhill due to gravity, the steeper the hill, the greater the gravitational energy to drive water down the river. That means that the fastest moving rivers are usually on the steepest slopes. The steeper the slope, the easier it is for the water to break down the rocks and cause high amounts of erosion. Therefore, fast moving rivers respond to their environment by causing increased amounts of erosion and sediment transportation to lower parts of the river. Likewise, if the slope changes, the river speed will change.

Based on the above, rivers change through time depending on their environment. If the slope changes, the river speed will change. If the material the river is flowing over changes (as in loose sediment, rocks, vegetation, etc.), that can also cause changes in flow speed, amount of erosion, and amount of sediment transportation. It is also common for rivers to flow into another stream, creating a larger river. This is dependent on how many higher streams are feeding into your river, and how many lower rivers your river will eventually flow into.