What goldsmith is pointing out is that in this kind of circuit you want the MOSFET to switch fully on or fully off without spending time in the transition between them. If fully off, no current flows so the the power (V*I) is V * 0 = 0, when fully on the current is high but the voltage is very low so the power is I * 0 = 0. If you slow down the rise and fall time by making the series resistor higher in value, the MOSFET spends longer in an intermediate state where both V and I are high and therefore power is lost as heat.
The diode PIV should be rated at the supply voltage plus a safety overhead, as a rule of thumb, use the highest rated diode you can in the space available. A voltage rating above what is actually there is not a problem but one with too low a voltage will potentially fail. The current can still be 1A. In your circuit the voltage across the diode reverses suddenly at switching rate (1KHz), a normal diode will not be able to start and stop conducting quickly enough and for a brief period as the polarity reverses, it will appear almost as a low value resistor and hence get hot. A Shottky diode (or 'fast recovery' diode) is engineered to minimize the time it takes to start and stop conducting so the 'hot' period is much shorter and overall heating much less.
Brian.