How often has a Test innings started with one captain bowling to the other?
And how many Test batters have converted their last two Test centuries into doubles?

Pat Cummins and Rohit Sharma were the 14th pair of captains to open an innings in direct opposition • Associated Press
You're right that India's chase in the World Test Championship final at The Oval at the weekend started with captain Pat Cummins bowling to skipper Rohit Sharma.
The two under-28s in the World Test Championship final at The Oval were the precocious pair of Shubman Gill, who's 23, and Cameron Green (24).
In the match you're talking about at Old Trafford in September 2020, Jason Roy and Joe Root were dismissed by the first two balls of the match, from Mitchell Starc. Even so, England looked like winning that game, before a sixth-wicket partnership of 212 between Alex Carey and Glenn Maxwell turned the tables.
You're right that Kumar Sangakkara's last two Test centuries were doubles - 221 against Pakistan in Galle in August 2014 and 203 vs New Zealand in Wellington in January 2015. But Ricky Ponting doesn't quite qualify for this list - the 40th of his 41 Test hundreds was 134 against India in Sydney in 2011-12 (he did make 221 not long afterwards, in Adelaide).
The longest duck in a Test match, by any measure, was achieved by the New Zealand fast bowler Geoff Allott against South Africa in Auckland in 1998-99. Allott lasted for 77 balls, and 101 minutes, before giving a catch off Jacques Kallis: he'd helped Chris Harris, who finished unbeaten on 68, add 32 for the last wicket.
Steven Lynch is the editor of the updated edition of Wisden on the Ashes