I'm not expert on this apparatus. But I'll try to give you logical reasons for your questions:
1) Assume you do not know the freezing point of your solvent. Even if you know it, that reference value is under different conditions such as atmospheric pressure (suppose you are in Denver of Mexico City, for instance). Thus, you need to determine the approximate freezing point of your solvent before you do careful reading. It's exactly like you are trying to determine the melting point of given compound. First time you set up MelTemp in high heating range to get an idea of melting range. Then you do careful heating with second capillary containing your solid to get melting point within $\mathrm {\pm 1 ^\circ C}$. The same approach applies here: First, direct heating to get approximate freezing point, then warm it to melt back, cool again directly to close to that temperature, wipe the outside of the tube, and this time, air jacketed it before do slow heating and get accurate reading. Whole this process is just to save your time and energy. That air jacket make long time to make your freezing tube with solvent cool (prime example: Dewar flask, with vacuum though). And, imagine if you have to stir that whole time with one stroke per second?
2) The latent heat of fusion release when a liquid undergoes liquid to solid transformation at its freezing point. If cooling is not uniform, the releasing energy may cause some solid near by to melt again (remember, at freezing or melting point, $\ce{liquid <=> solid}$). That's why frequent stirring is important, to give uniform cooling for this case.
3) You are correct about changing the instruction (c), but it should have been, "The temperature of the cooling bath should not be more than 4° to 5° below the freezing point of the liquid" to prevent the supercooling. But it's also worth noting that the set point of cooling bath temperature is subjected to change by each addition of the solute, because of the freezing point depression of the solvent.
