Learn Finnish: ”Kolmas kerta”, sanoi meklari ja nosti nuijan. - "Third time," the auctioneer said and raised the gavel.

Because of the direct quotation at the start of the sentence. When a reporting clause comes immediately after a quote, Finnish uses Subject-Verb inversion, putting the verb first (sanoi) followed by the subject (meklari). This functions exactly like the English literary style 'said the auctioneer'.

Why is the object nuijan in the genitive-like accusative case (-n) instead of the partitive (nuijaa)?

Because raising the gavel is a complete, finished action. When a sentence describes an action that gets fully completed, its object takes the total object case (here, the singular accusative ending in -n). If it were in the partitive, it would imply the auctioneer was in the middle of raising it or didn't complete the motion.

Since the subject and verb are inverted in the first clause, why isn't a subject explicitly stated in the second clause (ja nosti nuijan)?

Even with the inverted word order (sanoi meklari), meklari is firmly established as the subject of the sentence. Because the actor hasn't changed, the subject simply carries over across the conjunction ja. You don't need to repeat the noun or add a pronoun like hän before nosti.