Variable voltage supply (0V , 5V, 7V, 10V, 16V) rotary switch
we only need to use discrete components


Then presumably you can use NPN power transistors to make the zener regulator capable of delivering more power? I guess so! Here's the general idea: -



The unregulated supply keeps the 5V6 zener "topped-up" with enough current for it to regulate at 5V6. The BJT emitter follower is so-called because the emitter follows what the base does (apart from the loss of about 0.6 volts). This means that if the zener is 5V6, the output is 5.0 volts. If more current is drawn by the load this fractionally increases the voltage across the B-E junction and significantly, this turns-on the BJT to almost counteract the loading effect of the resistor load.

It's not perfect but it overcomes the loading problem you have with zener diodes. 

As an idea about this zener loading effect consider if you were trying to produce 5V with a 5V zener and you wanted to draw up to 500mA. The input supply is 12 volts and therefore the resistor R1 (14 ohms) drops 7V at +500mA (under no-load or full-load). This means that resistor is always dissipating 3.5 watts. Also the zener when off-load is dissipating 5V x 500mA = 2.5 watts.

The transistor circuit above is slightly worse at voltage regulating (compared to the zener) but, importantly, off-load it isn't throwing away power. Plus it's probably easier to get a suitable BJT in a T0-220 package than a zener. Note also that the zener and R1 in the transistor circuit only need to be low power devices - the zener voltage is being buffered by the BJT hence it only needs to be a 250mW device and the resistor probably rated no more than the same.

Many thanks to Tom Carpenter (who was also thinking along the same lines as me) but spent too much time (LOL) drawing the picture below: -



This would be the finished idea and has the benefit that you could select the zeners with common-emitter BJTs (digitally controlled) because they are only going to be consuming a few mA each.

As for the choice of transformer, this cannot be decided without knowing the full load currents you expect to deliver to your load at all the different voltage settings.