Minimum diode current / realtime clock battery backup
No, there's no minimum current for a diode. They're often rated at several mA, like for instance 10 mA, but that's because most users will use them in that range and then it's good to know what the forward voltage is.  



At very low currents the voltage drop will also be low, to less than 100 mV even; the graph for the BAT54 says 150 mV at 100 &micro;A and 25 °C. Either way, the DS1347 will operate at voltages between 2 V and 5 V so you won't have to worry too much about the voltage drop.