The device inside the MOC3023 is an opto-triac, not a diac. A diac is triggered by voltage across it, whereas the opto-triac is triggered by light from the LED.
The maximum current in the opto-triac should not exceed 50mA (except very briefly) since above that the triac is guaranteed to turn in in the relevant quadrants (I/III).
I am not familiar with your simulator, but in general for SPICE-based simulation you must have a ground reference in all subcircuits for simulation to work. The opto provides isolation so the circuit on the right is floating unless there are some fudges in the simulator or models. You can try grounding one side of the mains (for simulation purposes only) through a resistor such as 100K.
I don't know where you got the 200 ohm value from- that will allow almost 120mA of LED current to flow, far in excess of the 50mA absolute maximum. It only takes 5mA to trigger at 25 degrees C- check your calculations. 10 or 15mA nominal would allow it to trigger even in a Siberian winter, and after years of aging.
The only purpose of R2 is to limit the peak opto-triac current for the few tens of microseconds while the main triac is turning on. At a peak mains voltage of 311 volts that will be about 1.7A, then will rapidly drop to about zero. If you disconnect the triac MT2 the resistor will, of course, go up in flames. 180 ohms is okay.
One thing that may be confusing you is that the opto datasheet you linked has at least one error - the Itsm Y axis in Fig 4 (what determines the minimum value of R2) should be amperes, not mA! The maximum value is limited by how much voltage drop you can tolerate. If the opto triac and triac gates drop 2.5V total, the triac won't turn on until the mains exceeds Igt*R2 + 2.5V in magnitude or +/- 11.5V for 180 ohms. Too high and you limit the maximum output power, introduce asymmetry into the output voltage, cause unnecessary EMI at 100% power, and unnecessary heating in R2, so it is a trade-off.