Tributing “The Agent of Judgment - Saturn” is a cost.
You cannot activate “The Agent of Judgment - Saturn”’s effect during Main Phase 2. You CAN activate “The Agent of Judgment - Saturn”’s effect during Main Phase 1 even if you have to skip your Battle Phase because of a card effect.
You can activate multiple “The Agent of Judgment - Saturns”’ effects in the same turn.
You can chain “Barrel Behind the Door” to “The Agent of Judgment - Saturn”’s effect.
If you chain “Mystical Space Typhoon” to “The Agent of Judgment - Saturn”’s effect and destroy “The Sanctuary in the Sky”, “The Agent of Judgment - Saturn”’s effect disappears but it has already been Tributed so it is sent to the Graveyard.
If “The Sanctuary in the Sky” is not on the field, you can Tribute “The Agent of Judgment - Saturn” and activate the effect, but the effect is not applied and disappears.
If an effect is chained to “The Agent of Judgment - Saturn”’s effect that would increase your opponent’s Life Points, so that your opponent’s Life Points are higher than yours when its effect resolves, then “The Agent of Judgment - Saturn”’s effect disappears because your Life Points need to be higher than your opponent’s when the effect resolves.
Damage is calculated and inflicted when the effect resolves, so calculate the exact damage when resolving the effect, including any chained effects that altered the Life Points since activation; if the Life Points have changed since activation, the damage inflicted changes accordingly.