What is the typical consistency of a roux? When making a roux, how would you describe the final consistency that it should be after the butter coats the flour?  Should it be a lighter dirt-brown paste with a glistening buttery sheen, or should it be more runny/liquid?

Also, when melting the butter, is it best to melt it at extremely low heat and then increase the heat to medium after the flour has completely been incorporated into the flour, or can the butter be "rapidly melted" on medium heat while incorporating the flour?
Roux is very forgiving, and can be made a variety of consistencies, or ratios of butter to flour.  It should only be brown if you are intentionally making a brown roux, for the flavor.  Otherwise, it should be fairly yellow, closer to the color of the butter.

At the canonical ratio of 1:1 butter to flour (by volume), the consistency will be thick enough not to flow, but certainly not as thick as wet sand.  (When refrigerator cold, if you store it, it will be quite firm and thick, scoopable with a spoon with some effort.)

You can heat the butter at any speed you like consistent with not burning it.  You can heat it at fairly high heat until the foaming begins to subside, which indicates that the water has been boiled out.

I generally recommend adding the flour after the butter is melted, and stopped bubbling (which indicates the water is gone.)