You want to connect RGBI to the gate of the MOSFET (where you have RGB+ connected now). This is your control signal. You do not need to connect RGB+ - your strips are already getting power. You need to connect the rightmost pin of the MOSFETs to ground.
The Source of the MOSFET is the GND. Coming off the 12v?
All your grounds should be connected together (12v supply, Pi & TLC59711).
You should not need the resistor.
When the PWM signal line goes high on the Gate, the MOSFET will conduct current between the Source and the Drain. This will connect your LEDs to ground, completing the circuit from the positive side of the 12v supply.
Yes. The logic is inverted. The TLC59711 turns on LEDs by pulling the output low. But the MOSFET turns them on when the gate is driven high. Since you are using the TLC59711 signal to drive the MOSFET, everything is inverted.
Unfortunately, the flashing is a side effect of the way the PWM loads new data you send it. Essentially, it turns the PWM output off while loading the new PWM value. Since your logic is inverted, it shows up as a while flash.