Could sticking swollen finger in ice take swollen down?

Yes if acute injury: A swollen finger after an injury can be helped by ice. However, if swelling persists or if is late or prolonged one should check the cause for it as in infection, fracture, tendinitis. Here it is more important to treat the cause for the swelling than the swelling itself.

Often: It often helps, depending on the cause of the swelling. It is especially helpful with swelling caused by trauma.

Ice: I've is allways appropriate for swelling. No more than 15 min. Ice should not be directly on skin