Is it safe to do this experiment in the open lab?

Producing the equilibrium mixture in the U-tube is best done in a fume cupboard because of the toxic nature of chlorine. Once the tube has been sealed at both ends using three-way taps it can be brought out into the lab.

Will the equilibrium mixture persist for some time so that I can show the demonstration to several classes?

Yes. As long as the mixture is contained by three-way taps to prevent leaks it will last long enough to be used with a number of classes.

Could I show the effect of temperature on equilibrium position using different equilibria?

Yes. There are several other equilibria that lend themselves to showing the effect of temperature. The equilibrium between chloro and aqueous complexes of cobalt works well because the reactants and products are a different colour. You could also use the gaseous equilibrium beteen NO₂ and N₂O₄. You can produce the equilibrium mixture in a gas syringe and move the syringe betwen large beakers containing hot and cold water.

Are there other reduced-scale experiments that I could use in other areas of chemistry?

Yes, there are lots of different reduced scale experiments that you could use. The microchemuk website is a good place to start.