While this SIGNI is on the field, when one of your <Devil>SIGNI's power reaches 0 and would be banished as a game rule, if that <Devil> SIGNI has a Charm, you may remove the Charm instead. However, since that SIGNI's power is still 0, it is still immediately banished.
This SIGNI's [Constant] will also apply to your SIGNI that would be banished as a result of a battle.
When this SIGNI's [Constant] protects your SIGNI from being banished in battle against the opponent's SIGNI that has Lancer, your opponent will still crush 1 of your Life Cloth due to Lancer.
Anna Mirage, Devil Princess's [Constant] ability states "When one of your <Devil>SIGNI is banished..." meaning her ability will only trigger at the point of which a SIGNI is banished - in other words the actuality of the banish happening.
Therefore, because Paimon, Fallen Nihilism's [Constant] ability prevents banishes, Anna Mirage's ability will not work.
While this SIGNI is on the field and you banish one of your <Devil>SIGNI with a Charm with the ability of Death by Death, Death by Death's condition is fulfilled and one of your opponent's SIGNI loses -8000 Power.
Death by Death's ability states "Banish one of your SIGNI. If you do..."' meaning its ability will trigger at the point of which you (tried to) do the action of banishing your SIGNI.
Therefore, Paimon, Fallen Nihilism's [Constant] ability will prevent the banish of Death by Death, but not the condition to trigger its ability, because you did do the action of banishing your SIGNI, regardless of its failure.
Alternatively, Paimon's [Constant] ability substitutes the banishing of a SIGNI for the ability of Death by Death. Because you "did" the action of banishing, Death by Death's condition is fulfilled.