Question: Has anyone bowled a maiden in their only ODI over?
Spider says: Two people have (so far) done this, and the fact that both cases happened in (in ODI terms) ancient history leads to the conclusion that it is possible they will remain the only cases. Glenn Turner was New Zealand's first truly outstanding ODI batsman (he played 41 times between 1972/73 and 1983), though he wasn't much of a bowler. But after making 140 which played the biggest part in setting the Sri Lankans an implausible 305 in the Third ODI at Eden Park in 1982/83, his captain, Geoff Howarth, decided to give him an over (it appears to have been the last of Sri Lanka's innings, though no widely available records show this), and with his gentle fingerspin sent down an over accurate enough to be blocked out entirely. Earlier that same season, and against the same opponents, the rather less well-known Ashok Malhotra (who played 20 times for India between 1981/82 and 1985/86) was called on to bowl in his 3rd game, at Amritsar - and the circumstances were almost identical, as the Lankans had long since given-up hope chasing 270 in a match reduced to 46 overs per side, with Malhotra given what appears to have been the match's final over.
Q: And if so (or even if not) does anyone have an economy-rate of 0-an-over who has bowled more than a single over?
A :No - the only other man to have bowled in a ODI and not conceded a run is Australian opening batsman Walter Edwards, who played his only ODI in 1974/75, the one-off game which accompanied the brutal Ashes series. England won it comfortably, and with them needing a single to win Edwards was brought on and off his opening delivery England won the game. Whether they did this by scampering a leg-bye or courtesy of a wide or no-ball (in those days such extras were not debited against the bowler) is not recorded for public consumption.
Has there been an instance when a player batted or bowled on all 4/5 days of a match?
There have been many - one such example here
from Ian Botham in just his 4th Test match. What is much more unusual is players being on the field throughout a match.
Q: In ODIs, who has the best economy-rate of those who have bowled 100 overs?
A: England seamer Geoff Arnold, who played 14 ODIs between 1972 and 1975 (all bar a couple of which came at home, and one in which he did not bowl in as the game was rain-ruined), conceded just 2.84-an-over. He is one of only three specialist bowlers to have enjoyed a ODI career of any length whatsoever to concede less than 3-an-over - the others are countryman John Snow (9 matches, 1 in which he did not bowl, conceding 2.58-an-over) and Barbadian Sylvester Clarke (who played 10, 1 of which came for what was effectively West Indies A during the Packer schism, conceding 2.8-an-over).