Since gaining Channel Energy from different sources don't stack but basically give you two "pools" of channeling, how do other abilities that modify channel energy work? For example, if I take a level in Cleric and take the Sun Domain which says, "Undead do not add their channel resistance to their saves when you channel positive energy." and then I take enough levels in, say, Paladin, do get channel energy also, would the Sun Domain text apply only when I used my Cleric pool of channels and not my Paladin pool? Same question for feats, would I have to choose which pool the, say, Turn Undead feat applies to?
Sure thing. Let's look at Sun's Blessing:
Sun’s Blessing (Su): Whenever you channel positive energy to harm undead creatures, add your cleric level to the damage dealt. Undead do not add their channel resistance to their saves when you channel positive energy.
The text states what the ability allows the creature to do. In this case, the extra damage applies "[w]henever you channel positive energy to harm undead creatures". There exists no rules text, specific to this ability or general to class features that reference other abilities, limiting the scope of this text to the class that granted it. Any other rules-based limitations on the ability would have to reference some other rule. (For example, you can't use it while dazed because the rules for the dazed condition prevent you from taking actions.)
Edit for elaboration:
To get a feel for this kind of thing, one can take a look at other situations like it. The first thing that comes to mind is sorcerer bloodline arcana that augment or otherwise alter spells being cast, such as the elemental bloodline. When this question came up regarding, e.g., a multiclass sorcerer/cleric casting fire storm from the cleric list, Paizo issued a FAQ confirming the plain reading of the text--"[w]henever you cast a spell" means any spell, not simply a spell provided by the class that gave you the ability.
So in at least that case, the plain reading of the text matches the stated intent of the rules team. When confronted with a similar situation, it makes sense to decide in a similar fashion--unless there is a significant difference, such as other rules text or another FAQ or developer commentary, that would affect the evaluation.
i think this may apply https://paizo.com/paizo/faq/v5748nruor1fm#v5748eaic9ne8
Sorcerer: Do the bonuses granted from Bloodline Arcana apply to all of the spells cast by the sorcerer, or just those cast from the sorcerer's spell list?
The Bloodline Arcana powers apply to all of the spells cast by characters of that bloodline, not just those cast using the sorcerer's spell slots.
General rule: If a class ability modifies your spellcasting, it applies to your spells from all classes, not just spells from the class that grants the ability. (The exception is if the class ability specifically says it only applies to spells from that class.)