If the light is flickering it's absolutely guaranteed that it's derived from this.
Not necessarily. If it's a 50/60 or 100/120 Hz flicker then yes, but LED lights with a cheap switching power supply might still flicker at the switching frequency, which could be say 400Hz or something.
You have to be intentionally wasting money to put in a transformer big enough to handle 400Hz, and once you get into the lots of KHz where a supply like that is happy I think your "please don't explode" capacitor on the transformer is enough to prevent flicker.
Not necessarily. If it's a 50/60 or 100/120 Hz flicker then yes, but LED lights with a cheap switching power supply might still flicker at the switching frequency, which could be say 400Hz or something.