The two most common training aircraft out there are the Piper Cherokee (numerous variants) and the Cessna 172/152 (numerous variants as well). The cherokee is not spin certified and the 172 is only spin certified in the utility category so I would not try spinning either frankly. Technically you may be able to recover from a spin with enough altitude and depending on the situation but a pilot could get them selves into a nasty situation by spinning either plane. Even regular stalls put you at risk of entering a spin.
There are lots of maneuvers (aerobatic mainly) that would kill the engine on many common GA planes. Generally speaking the oil systems are not built for inverted maneuvers or anything that would starve the oil pump. The Cessna also has a gravity fed fuel system which would not do to well inverted. In regards to stalling the engine and recovering while you may be able to glide down to a field if an engine is lost this is more in reference to losing an engine during a maneuver and having insufficient thrust to recover properly from the maneuver.
I would think that you might also be able to get the plane into an aggressive dive that might be hard to recover from for an untrained pilot. At some point the recovery will exceed the load the wings can handle as well.
Here in the US the FAA requires knowledge of spin awareness and what may cause a plane to enter a spin but there is no requirement to demonstrate a spin for a basic private pilots license nor is there a requirement for them to be demonstrated to the student.