What is the precise role of the control in this experiment?
The two sets of equipment were chosen to be identical. However it is possible that some differences were overlooked, maybe the two funnels are made of different glasses with different insulating properties or one of the set ups was sitting in bright sunlight whilst the other was shaded. By switching the two over any possible bias can be removed.
Why should you wait until the ice has begun to melt before turning on the power and starting the clock?
The ice would have been made in a freezer or some kind. The temperature of the freezer, and so the temperature of the ice when it was removed, would be below 0 °C. By waiting for the ice to start melting you can be confident that the bulk of the ice is at or very close to 0 °C. Without this step additional energy from the heater would be needed to raise the temperature of the ice to 0 °C. This would over estimate the energy needed to melt the ice and so over estimate of the specific latent heat of fusion.
Why are (both) set ups allowed to stand for a further minute before they are weighed?
When the clock was started the heater element was cold, as a result the first few ten seconds or so were spent heating up the heater. When the power is turned off the heater is still hot. Some of the energy that has been delivered to the heater, and recorded as having been, will still be in the body of the heater. Allowing a minute or so allows the heater to return to the temperature of the ice (0 °C), and so all the energy delivered to the heater has been passed on to the ice. Failure to do this will lead to an over estimate of the energy that has been delivered to the ice and so an over estimate of the specific latent heat of fusion.
Why should the experiment be repeated with the principal and control setups reversed?
The control is required because heat does not only enter the ice from the heater. The main source of heat is by conduction through the funnel walls but the metal body of the immersion heater might also carry some heat to the top of the ice heap (most heat from the heater element will be delivered close to the end, at the bottom of the funnel). Without the control these additional sources of heat would add to the figure for the mass of ice melted and so underestimate the figure for the specific latent heat of fusion