The entry for Dying God in my Pet Devotion guide has the relevant details, as quoted below:
Given the above, your questions are pretty straightforward:
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Yes, health drain applies only to the player. In fact, as with other pet buffs, Hungering Void cannot be bound to pets.
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As you can see from the list of bonuses above, there’s a lot more than just vitality or chaos damage bonuses for pets. The guide explains why the most important bonus on the list is pet crit.
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It’s better to take Dying God and learn how to cope with its drawbacks. Per the guide, this is the only devotion with a drawback like this, but there are many ways of “diverting resources” to help address it: via devotions, items, mastery skills, etc. The main thing to keep in mind is that unlike other devotions, you need to invest in additional sustain as part of the cost of the devotion, and this can be especially tricky for pet builds since they don’t have easy use of weapon-based ADCTH. Because of this, Dying God is easily the most expensive devotion of them all, and that’s why it has such an impressive list of pet bonuses.
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Healing Rain would be one example of the additional investment needed for Dying God, but it’s one of the more expensive solutions as well. If you look at the various builds that use Dying God, you’ll see lots of different solutions used.
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Generally, offense is more valuable defense because it’s quite feasible to make an immortal character in the game, but that character wouldn’t be able to clear much content, which makes it pretty useless. Also, having a high enough offense decreases the need for defense by shortening fight times and reducing the risk of being overrun. The trick is to give up as little offense as possible to reach just “enough” defense, with “enough” being defined by whatever your goals are, such as beating Ultimate, soloing Gladiator, etc.
Good luck!