Will the Antichrist, upon his emergence, claim righteousness first, then prophethood, then divinity, or will he claim divinity directly, especially since there are hadiths that indicate both matters?

The Dajjal will not claim divinity directly. Rather, he will first claim righteousness and faith, then prophethood, then godhood. This is as mentioned by Al-Tabarani, Al-Saffarini, and Ibn Uthaymeen, and it does not contradict the Hadith of Al-Jassasah.