The steady state is reached when the rate at which heat is entering the rod is exactly equal to the rate at which heat is leaving the rod. The temperature of the rod is independent of the rate at which heat is flowing through the rod.
Think of it like a bucket with holes in it that is being filled from a tap. Once the rate at which water leaves the bucket equals the rate at which water enters the bucket, the height of the water in the bucket will remain the same - this is the steady state. If the bucket has many holes then height of the water in the steady state may be very low. If the bucket has only a few holes then the height of the water in the steady state will be higher - the bucket can hold more water if it is less leaky.
You also ask how different parts of the rod can have different steady state temperatures. Well, you can think of the rod as being made up of lots of little rods placed end to end. The steady state temperature of each little rod will depend on the temperatures of the rods on either side of it, which need not be the same. So each little piece of the rod may have a different steady state temperature, even though they all have the same amount of heat passing through them.
In the limit, as you make each little rod smaller and smaller, you end up with an equation that tells you how the temperature at each point in the rod depends on the temperatures on either side of that point - this is a partial differential equation called the heat equation. Solving this equation then tells you how the temperature at each point in the changes over time, and what the steady state temperature is at each point along the rod (if there is a steady state).