Does a transistor/FET dissipate more power with a freewheel diode?
If we assume the diode has 2pF capacitance and 1nA leakage when reversed, then yes. The transistor will have a transient slightly larger current when switching on, due to charging the diode capacitance, this can be neglected as it only occurs once per operation. There is a continuing slightly larger current due to the diode leakage. You do the sums on the relative influence of the continuing extra 1nA, and decide whether it matters or not compared to the relay current.

The major difference comes when the transistor switches off. With the freewheel diode, the relay kickback (that is the continuing current driven by the energy stored in the relay coil inductance) is conducted through the diode, and the collector voltage rise limited to less than one volt above the rail. Without the diode, the kickback raises the collector voltage until the collector-base junction punches through, destroying the transistor.