Can a power mosfet switch small loads?
I couldn't find any 'minimum ID' parameter

Look up \$\color{red}{I_{DSS}}\$ in the data sheet of the device - this tells you what the leakage current of the MOSFET is when it's turned off. Clearly, to be effective as a switch you don't want \$\color{red}{I_{DSS}}\$ to encroach any appreciable distance towards your normal operating current AND, by the same token you don't want it to semi-power the load when it's meant to be inactive.
Typical values of \$\color{red}{I_{DSS}}\$ range from 100 nA (really good) through 1 uA to 250 uA (not so good for light loads). An IRFZ44 (a pretty commonly used device) is 25 uA but this can rise to 250 uA at higher temperatures.
If your load is very light, you'll probably do better using an analogue switch to turn your load on and off because these devices can be rated down to leakage currents of less than 10 nA with an on-resistance of 1 ohm. Horses for courses.