So, he worked for the Russian state until 2020, then? Or until March 2022, when he resigned as head of Skolkovo?

As you can see, he was sticking around until the war actually started. Did he precipitate the war by lending his name and skills to Putin? IMO, definitely yes.

Should he resign from FIDE? Russians do not resign, they cling to positions of power indefinitely.

Should FIDE fire him? Definitely yes, its rules are made for respectable gentlemen.