You can find all answers to your questions on official ACO Le Mans page at Retrospective/Roll Of Honour And Records/Dates and timetable of the race.
Dates
Always in mid-June (summer solstice) except in:
1923 : 26-27 May (first race).
1956 : 28-29 July (delayed by construction work).
1968 : 28-29 September (delayed by the events of May ’68)
1986 : 31 May-1st June (clash in motor sport calendar).
Horaires
Traditionally from 16:00 Saturday to 16:00 Sunday, except in:
1968 : Start at 15:00 (race put back to September).
1969 : Start at 14:00 (presidential election).
1979, 1981, 1984, 1985, 1988, 1996 : Start at 15:00 (elections except for 1985, European Cup Football in 1996).
1998 : Start at 14:00 (TV coverage shared with the tennis final at Roland Garros and football World Cup)