MOSFET push pull gate driver: Too high base voltage?
The circuit makes perfect sense.

Now answering your questions:


Yes, Vbase maximum is 6 V, but that is VBE, and you are measuring from base to GND, which is here Vbc, right? Please note that BJT is a current-controlled device, so adjust Rbase to achieve the gain based on the Ib/Ic curve (beta gain).
Q1 is causing this to happen, since when MCU is high, Q1 turns on driving the totem pole (or emitter follower pair) to Gnd, so the PNP Q2 will conduct. That will drive the MOSFET gate to GND, turning it off. So at this point the LEDs are just connected to Vcc and they will be OFF, and vice versa!