I don't know the exact name of this trail. The mountain group in the background is called Cadini di Misurina. So the riding is between there and Drei Zinnen / Tre Cime di Lavaredo. South of Rifugio Auronzo, around 20 minutes from it. Not really a secret spot... Stunning scenery, but from a mountainbiking perspective probably not worth coming from the US to Italy just for this ;-)
Hard to tell. Judging from Photos and Photo Spheres on Google there can be a lot of snow until May. I can not find any shots from June. From July on it seems to be snow free every year. You might ask a local.
Been in that general area quite a few times over the last couple of years, for hiking trips mostly though. Last year we had to wade through knee-deep snow in the region above 2500m when hiking up to Tschigat in late May.