It’s a participle being used as an adjective (like “the walking dead” or “the setting sun”). It’s a little unusual, in that the phrase “next customer” is the common idiom in English, but I can’t see what could possibly be wrong about it.
Both terms are fine; neither is incorrect, and in English they refer to the same phenomena. Genitive has a checkered past as a case name, but English no longer has cases, so possessive carries fewer Latinate presuppositions. The doctor is clearly a noun phrase, and the clitic -'s marks it as a possessive. – John Lawler