The engine conversion is possible, and yes they can use the original tranny.

They would not beable to install the original ecu from the '99 civic, however the original ecu could easily be remapped to run the '99 D16y7 map.

So put simply...
If the map has been changed to the '99 map on the original ecu, then it will run perfectly fine.

If the car is still running the D15 map(standard ecu not modified), it would run but i would be concerned that the engine is running safe (correct air/fuel ratio)