They are involved for the technical aspects beyond merely MachineGames using id engines. MachineGames in a lot of ways acts like another id software studio, but it is free to form its own flavor.

MachineGames also has (uncredited?) work on Doom 2016 and Doom Eternal.

Wolfenstein was made by Machinegames. Though there is quite a bit of overlap between the two studios during production.

2009 - they were producers, but the development was done by Raven Software and published by Activision. So not really involved other than owners of IP?

New Order - not at all, maybe as engine developers. Right for that IP were transferred to MachineGames in 2010 right after ZeniMax got hold of them.

As for Doom 2016 - they enlisted a lot of outside help after Doom 4 was scraped. Bethesda's game directors helped them a lot because they already figured how to make "old ip" to sell well with modern gamers (see Fallout 3).

side note:

I don't think id managed to get deliver a lot of good games since John Romero left. (just like John Romero didn't deliver many good games since the separation)

John Romero and John Carmack were like a dream team, but without each other it was meh.