I get ambulance cases that turn out to be a sprain or some other minor injury. My docs diagnose that and slap on the "numbing ointment". Then they lie there taking up a valuable bed while new ambulance cases wait in the rain.

Are there any better possibilities of transferring these patients elsewhere other than the HDU ward of another department or the ICU? AFAIK the game does not let you move them to observation (in Emergency) or the regular wards of other departments.

If the house icon doesn't appear on the patient sheet then they have other injuries that need tending to before they can be dismissed. If they're given IV fluids or something else that takes a long time, they won't be eligible for dismissal until the next morning (somewhere between 6:00 and 8:00, I forget specifically when). The best solution is to transfer them to a department's Regular Ward. If you just want to clear the Trauma Center bed, an alternative is to send them to the ICU. Early game, you tend not to have to use the ICU much but it literally accepts all patients. As such, it makes for a great patient overflow.