G'day BravoV, some answers for you
Q: At the on state (switched ON), once the relay is fired, will the triac off instantly ?
Yes, when the relay contacts close they take the current from the triac and it goes non-conducting. If the contacts open with current flowing thru them, say during a bounce, an arc will be struck between them. The voltage across the arc shouldn't be more than 20 V so the triac is not likely to be retriggered and even if it is triggered, the contacts will close again shortly.
Q: So the arc on the relay contacts will still happen even there is a U3 MOV (390V) across it ?
I was referring to the switch contacts opening, not the relay contacts as the relay contacts should never open with current thru them. The voltage needed to strike an arc between contacts that are opening is on the order of a few volts and a MOV is practically useless. The voltage is so low because when contacts are opening electrons can jump the gap when it's still much less fractions of an mm. If enough current is flowing the electrons vaporize the contact metals making them conductive ions and forming an arc. An RC snubber is much better because if sized properly it can carry some of the current and extinguish the arc.
Q3: Ok, I'm abit lost, so there is a chance that this circuit won't turn off ?
No, it will shut off because the input is AC and the instantaneous current will sooner or later drop to zero after the mains switch is opened. The the arcs extinguish and the voltage isn't high enough to re-establish them. Good thing this isn't a DC system where arc can and does go on, and on, and on...
I'm referring to the case where the switch happens to be on but the triac hasn't yet been triggered. If a voltage spike comes down the line at just the right time, the triac could be triggered too soon and orginal problem occurs, ie. the transformer core saturates and the fuse blows or the breaker trips. This shouldn't happen often but since you are using this in an electrical lab, you could very well have voltage spikes on the mains that a snubber could help with.
If you find this still happens, then a slightly more complicated trigger circuit will be needed. I would exchange D1 & R1 then add provisions for a capacitor between their junction and the cathode of U1 (right hand side). This will form an RC network to attenuate any spikes from the line. I'd probably remove snubber A. R1 would then be something like 470, 1/2 W and the new capacitor 100n, 630 V. I'd also move U3's right hand terminal to the net connecting pin 4 of the power switch.
Hope this is a bit clearer,
Take care,