How to connect IRLZ44N to Arduino?
It's better with resistor so output driver isn't stressed to its maximum limits. Gate has its capacitance so it's better to limit charge/discharge current for such weak source
As the gate is basically capacitor, it has really big input impedance. Without pull down resistor it'll be catching all kinds of EMI (electro-magnetic interferences). So 10k is fine. The 100k might be less unreliable if the mcu output is not active (boot/reset)