1. Is quick opening and closing of the furnace doors, given the small height of the workpieces, a good idea?
This is a good idea. You could also consider quick-opening doors into entrance and exit vestibules (cold chambers) at each end of the furnace to reduce heat loss and minimize atmosphere disruption.
2. How about using a preheated zone before and a heated zone after the working zone of the furnace, with internal furnace doors? (This is an expensive option, I presume.)
Expensive and possibly high maintenance, but it's an excellent method of reducing heat loss and minimizing atmosphere disruption.
3. What about using a hot-air curtain (of course, only when the furnace door is open)?
This is not a good idea. the concern is air/gas infiltration into the furnace.
4. What about using a curtain made of high-temperature-resistant material in front of the furnace doors?
Metallic chain curtains or curtains made from materials like Rafasil (high-temperature ceramic-fiber material blend) have been used. They reduce but do not eliminate heat loss and must be replaced when worn to stay effective. The question is, will they be viable in a production environment.
5. Should we program the furnace so that the fan stops when the doors are open and more heat is supplied?
This is a bad idea. The starting torque to get the fan going again after sitting still in a hot chamber can damage the fan and/or significantly shorten fan life.