Looks reasonable enough.
Yes a triac can be on with no gate voltage, buit not for long on ac.
Once triggered the triac stays on untill the voltage across mt1 and mt2 goes through zero, then it turns itself off.
If you went for the moc3040 the datasheet tells you how to connect to an external triac, there might be a delay in switch on and switch off of the triac from the input signal, but it will only be 20mS max (1/2 mains cycle), I dont think this would be an issue for your design.
Triacs have diffo trigger currents for the 4 mains quadrants, so long as your gate resistor is correct this wont be an issue for this application, esp seeing as your load is resistive.