Once Mobius effect is placed on the chain, his presence on the field has no bearing on what happens to his effect anymore. Mobius effect is now Chain Link 1 on a resolving chain. Mobius is not getting his effect face-down, as no monster can activate an effect when the game doesn't "see" the card in play. Mobius is having his effect activated, placed on the chain, while he is face-up. Whatever effect chained to that trying to affect Mobbius will only affect the Monster Card and not the effect that was triggered off of it.
Once an an effect is placed on a chain, there is nothing you can do to prevent it from resolving, other then using another effect that negates it, or setting up conditions to where the effect cannot resolve properly. Book of Moon does neither of these.
Spirit Reaper is a completely different scenario. Reaper is a Continuous Effect, not Trigger nor Ignition. It is always on, and never uses the chain. The reason Reaper cannot destroy himself until the card targeting him resolves is because Continuous Effects can only insert themselves between links of a chain, but they can never interrupt a resolving link. His effect has to wait until an effect like Book of Moon resolves before his Continuous Effect can apply. Sine he is no longer face-up, his effect no longer applies.
Spirit Reaper's effect is continuous and does not use the chain. Mobius the frost monarch's effect is a trigger and does use the chain. Once an effect is placed on the chain, unless the text of the effect is specifically tying the effect to the card being face-up, then the effect is not dependent on the card being face-up.