Better to carefully cut just above the fruit. Pulling or twisting can injure the peel, tearing it, making the fruit rot faster.

To my knowledge, they need to be removed manually. Any citrus I've grown (potted, mind you), if allowed to hang on the tree until the fruit falls naturally, are already at the point that the fruit quality is going downhill, getting kind of dry and puffy inside.

I have about 30 citrus trees in the ground. The majority of them are tangerinextangelo hybrids with thin or soft skins. Last week I harvested two boxes of fruit by twisting the fruit and had only one torn plug.

If I only had one tree, maybe I would cut them. If you cut them you are risking the spread of disease just like in prunning, and should sterilize between trees. Usually there is not enough time to do that. And I ruin enough cloths with bleach when I have to prune so I leave my shears in the house.