If you are worried about the alkalinity, use a slightly acid wash to neutralize it. Put a little white vinegar, lemon juice, or equivalent in the wash water. Consider leaving the carpet at home. The runway is located in a firm, dry, smooth portion of the playa. If you are flying to the desert at other times, or visiting the far reaches of the playa, you won't have a runway.
Dark areas are wet and soft.
The mud gets a dry white glaze on the surface. The tan areas in the center of the playa are your best bet. You can make a soft-field landing on a firm surface. The reverse is not true. The edges of the playa get increasingly softer.