Power Mosfet design issue
1) Yes.

2) No. Your mistake was in not realizing that the PWM output from the MCU couldn't get close enough to ground to turn Q1 OFF all the way, and therefore turn Q2 OFF all the way.

3). Yes, and you shouldn't use a diode, you should use a resistor and it should be  something way less than a megohm in order to keep Q1's output transitions crispy. Depending on the MCU's supply voltage, 10k to 100k might be OK.