Racing occurs virtually every day of the year, save for Christmas--and there may be a track somewhere that runs on Christmas too! The southern circuits (Florida, Louisiana, So. California) run more during the winter, but New York still has winter meets at Aqueduct in NYC and West Virginia and Kentucky also run during the winter months.
Some specific tracks--I'm thinking of Mountaineer Park in Chester, WV in particular, race year around with one or two days per week "dark" (i.e. there is no racing).
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Depends on the length of the race and the quality of the horses. Most races last anywhere from 1 to 2 minutes.