If I wanted to go mountain biking in the village of Red Hook, would I find a good trail?

In the village of Red Hook, no. Outside the Town of Red Hook, yes. The only places I can think of where I know that'd be available is in the Taconic-Hereford Multiple Use Area down near Union Vale, or across the Hudson in the Catskills, which aren't all that far away. I'd personally think the Catskills are a more appealing target for mountain bikers, but your mileage may vary. There may be others nearer to town.