According to Azerbaijan and Turkey, in the ceasefire agreements of the 2020 war of Azerbaijan against Artsakh, Armenia had allow a land corridor to Nakhichevan, a so called Zangezur corridor. What is the status of the corridor now after a new war and a complete capitulation of Artsakh without any involvment of Armenia itself? Does Azerbaijan still demands this corridor even after taking over Artsakh? On which grounds?
I think it's a unclear at this point. The government of Armenia isn't saying much and appears rather paralyzed for the moment. In a repeat of what happened after past Azerbaijani battlefield successes, there have been mass demonstrations in Yerevan demanding the government's departure. The only thing rather concrete that PM Pashinian has said is that he blamed Russian peacekeepers for [allegedly] not doing their job in preventing the Azerbaijani offensive. (IDK what the deals/treaties entailed in this instance, but if they were like UN peacekeepers' mandates, it was probably not even their job to stop any Azerbaijani offensive.)
Does Azerbaijan still demands this corridor even after taking over Artsakh? On which grounds?
Almost certainly. Not only do they demand it because it would link their lands, but Türkiye's government demands it due to their wider geopolitical projects.
As for "grounds", they e.g. say the area was given to Armenia by USSR only in the 1920s and that it's historically Azerbaijani land, etc.
And in case you think I'm inferring too much, here's Erodgan saying it is a "strategic issue" for both Azerbaijan and Türkiye:
OTOH, since Armenia is still opposed, both Turkish and Azerbaijani goverments seem to have de-emphasized that angle in recent statements and say instead it will go through Iran:
"If Armenia does not pave the way for [the corridor], where will it pass through? It will pass through Iran," he said. "Iran currently considers this positively. So, it would be possible to pass from Iran to Azerbaijan." At a cabinet meeting later in the day, Erdogan reiterated the statement.
Erdogan's argument has not been confirmed by Azerbaijani officials, but the idea was presaged in a September 17 article on Haqqin.az, a website associated with Azerbaijan's security services. It differentiated between a "Western Zangezur Corridor" through Armenia and an "Eastern Zangezur Corridor" through Iran. It concluded: "If Yerevan continues to delay the opening of the Western Zangezur Corridor, then Azerbaijan will open the Eastern Zangezur Corridor with Iran, which means that Armenia will remain outside of yet another strategic project and will once again be a loser."
As for what they are dangling in front of Armenia in exchange is a normalization of relations and opening of trade.
OTOH, according to some analysts (quoted in that RFERL piece), not insisting/threatening for now may simply be because they don't want Pashinian's government to fall right now, due to piled up internal political pressures. After all, he did give up Karabakh without a fight, so he's seen from Baku as a more reasonable person in charge than alternatives might be. But as far as the corridor goes, their position appears unchanged:
"Armenia has never agreed and will never agree to any sort of extraterritorial or corridor logic," Armenian Territorial Administration and Infrastructure Minister Gnel Sanosian said on September 25.