Q: Is it true that Derek Underwood never hit a six in his Test career?
A :Incredibly, it is. Underwood played 86 Tests and never cleared the boundary once. Deadly Derek's skill with the willow was negligible and bats in his day were weak by modern standards, however. Lance Gibbs, who played 79 without hitting one, is not too far behind. And Glenn McGrath had played 101 before hitting his first any only six as well. This came in the same match he made his only Test half-century.
Q: I know Warwick Armstrong holds the record for most Tests as captain without defeat, but who comes second?
A: A pair of England captains are tied in second place: Brian Close briefly held the post in 1966 and 1967, leading 7 times without defeat, to equal George Mann's record from nearly two decades earlier. Mann's entire Test career came as captain, and he never lost.
Q: Has any wicketkeeper ever failed to take a catch in their career?
A: Yes. England's Don Brennan played two games, in fact, standing in for Godfrey Evans in 1951. He made one stumping but took no catches. There are five others (including Humayun Farhat, who may possibly play again, though this appears unlikely) who played a single game without a catch.
Q: What's the most Test innings' played without scoring a duck? And who was the player concerned?
A: Australian all-rounder Jim Burke (perhaps better known for a suspect bowling action) played a fairly remarkable 44 innings' and was never dismissed without getting off the mark. This is perhaps even more surprising as Burke was a notorious slow scorer and would have inevitably spent many deliveries on zero.
Q: Who has the best batting average for a single Test match: i.e. who scored the most runs for either one or two dismissals? (Presumably one)
A: Mark Taylor, in a somewhat notorious game at Peshawar in 1998/99, scored 334* and 92, for a match total of 426 for once out. This trumped Brian Lara's 375 in his single innings at the ARG in 1994.
Q: Who waited longest in terms of matches to score a century?
A: Depending on whether you mean the longest wait from debut to maiden century or the most consecutive games between centuries, the answer is Shaun Pollock (whose first came in his 51st game - and he almost immediately scored another) or Adam Parore, who played 56 games between his two Test centuries.