That innings of 201 by Greg Chappell for Australia against Pakistan in Brisbane in 1981-82 did indeed include 177 runs apart from the 24 in boundaries. That's fairly high on the list - but pride of place goes to Pakistan's Hanif Mohammad, who ran 241 of his 337 (there were 24 fours) against West Indies in Bridgetown in 1957-58.
The answer to your particular query is the second Test between England and Bangladesh in Chester-le-Street in 2005 - it was all done and dusted in 1144 balls (190.4 overs), with England winning by an innings after just 3.5 overs on the third morning. The previous Test, at Lord's, was actually completed three balls quicker, but featured only 795 runs, so it just falls short of your qualification.
There have now been 23 instances of a batter being stumped in both innings of a men's Test. The first to suffer this fate was the England captain "Monkey" Hornby, against Australia at Old Trafford in 1884. The most recent instance was by Sikandar Raza, for Zimbabwe against Sri Lanka in Harare in 2019-20.
The Sri Lanka fast bowler Lasith Malinga was on top of the IPL wicket-takers' list for quite a few years - he finished in 2019 with 170 - but he's been passed during this season's competition by Dwayne Bravo of West Indies. As I write, he has 181; he went past Malinga when he dismissed Deepak Hooda during Chennai Super Kings' match against Lucknow Super Giants in Mumbai on March 31.
The man saddled with this strange sobriquet was opener John Rutherford, the first Western Australia representative to play a Test; he won one cap, against India in Bombay (now Mumbai) in 1956-57. Sadly, he died last week at the age of 92.
Steven Lynch is the editor of the updated edition of Wisden on the Ashes