I'd wager that the turbojet only kicks in after the booster is disconnected. There are two reasons for this, a) in the FNA picture I posted above, you can see the missile immediately after the booster is released and there's no visible indicators that the engine is running, and b) if the turbojet engaged while the booster was still attached it would burn it off! I'd also assume as you do that the airflow is necessary for the missile to start, hence why the fixed-wing-launched versions don't need the boosters, but the helicopter-launched ones do.
You bring up a good point though, which is the failure rate of any of these missiles; I don't really have anything to add to the discussion beyond saying that the missiles are all probably well within their shelf-life, especially these new Qaders. I can't imagine the risk of engine failure to be that significant since Iran's chosen to make the Noor their primary ASCM as well as expanding its use in other platforms like the Karrar (which uses some variant of the Toloue), and the Qader.