Great question, although I quibble with your use of the word 'Core' instead of 'Basic'.
1. Should the Core require Cleric healing? Is it OK to have a party of four non-healers survive an adventure long enough to get loot and go home?
No, I don't believe the Basic game will require healing spells. Not having them in the party will just reduce the number of fights they will want to get into before stopping to rest. Will having a class with a healing spell help extend that? Sure. But it won't be required.
2. Should the Core require Cleric healing? Is it OK to have different classes (Warlords, Bards, etc) that also fulfill the required healing role?
Yes. The Basic game is Fighter, Rogue, Cleric, Wizard. That's all. If you want healing or recovery that comes from another class... play Standard.
3. Should the Core require Cleric healing? Is it OK to have different types of abilities that work similar mathematically to healing (temporary hit points, parrying, dodging, etc) and one of those be required?
I believe Temp HP, parrying, dodging etc. are all more advanced rules for melee combat that Basic is meant to support. They won't appear in the Basic rules and thus would not be a replacement for healing spells for faster-than-normal-resting-recovery. If you want them... take your Basic rules, add in the Fighter's Parry and Protect maneuvers (thereby instantly turning your Basic game into a Standard game) and you're good to go.
1. Should the Core require Cleric healing? Is it OK to have a party of four non-healers survive an adventure long enough to get loot and go home?
No, I don't believe the Basic game will require healing spells. Not having them in the party will just reduce the number of fights they will want to get into before stopping to rest. Will having a class with a healing spell help extend that? Sure. But it won't be required.
2. Should the Core require Cleric healing? Is it OK to have different classes (Warlords, Bards, etc) that also fulfill the required healing role?
Yes. The Basic game is Fighter, Rogue, Cleric, Wizard. That's all. If you want healing or recovery that comes from another class... play Standard.
3. Should the Core require Cleric healing? Is it OK to have different types of abilities that work similar mathematically to healing (temporary hit points, parrying, dodging, etc) and one of those be required?
I believe Temp HP, parrying, dodging etc. are all more advanced rules for melee combat that Basic is meant to support. They won't appear in the Basic rules and thus would not be a replacement for healing spells for faster-than-normal-resting-recovery. If you want them... take your Basic rules, add in the Fighter's Parry and Protect maneuvers (thereby instantly turning your Basic game into a Standard game) and you're good to go.
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