Control of a coreless DC motor using a PWM and a mosfet
Reduce the 10K resistor to 3.3K or even 1K.

The Si2302 device that you are using has threshold voltage around 0.65V, this is much lower than many MOSFETS which more typically have a a threshold of 2v - 4.5V.

All effects you are seeing are probably because of the pullup resistor in the Arduino (assuming it uses an AVR processor).  This can be anywhere between 20K and 100K and will be putting enough current out of the processor to create a bias voltage that turns on the MOSFET slightly.

When you measured the gate resistance, was it still connected to the Arduino?

The pull-up can be disabled programmatically, but of course not until the software is running.

kevin