This strange thing is growing on my jasmine there is only this one and I have never seen it before. It is firm to the touch and is attached just the same as a leaf.
Those are seed pods! Here are some articles that will offer more information on the plant:
It did flower when it was a small plant.
This information should help:
the yard for the fall & winter...? The winter gets pretty cold here.
Unless you know the variety name and whether it is hardy in your zone 7, I suggest taking it in for the winter. Some jasmines are not hardy in cold weather, however, common jasmine is hardy to USDA zone 7. Some need to be planted in zones 8 to 10.