#mana abilities

If any of these criteria aren't met, then the ability isn't a mana ability and uses the stack.

No, this isn't normally possible.When your opponent activates their Grove of the Burnwillows, you gain life as part of its mana ability resolving. This doesn't use the stack and can't be responded to. If your opponent is casting a spell or activating an ability, then Death's Shadow will die from state-based actions once that spell is cast or the ability is activated. If your opponent is just activating Grove of the Burnwillows while not in the middle of casting a spell or activating an ability, then state-based actions will be checked immediately.The only possible way that your Death's Shadow could survive here is if your opponent is paying mana into something that you can also pay mana into, such as a spell with Assist or Join Forces. In these rare cases, you can use your Ancient Tomb to reduce your life total before state-based actions are checked. A fetchland is never going to be able to help you here, though.

Yes, you can do this.Food Chain's ability is a mana ability, which means it can be activated while you're casting a creature spell. The point at which you activate mana abilities while casting a spell comes after you've determined that spell's total cost, and so the cost reduction effect from Krallenhorde Howler has already been applied and the total cost locked in. This means that sacrificing it now won't undo the cost reduction and so your creature can have its Howler and eat it too!

No, these abilities are mana abilities and they can't be responded to. Any activated ability that could produce mana, doesn't have a target, and isn't a loyalty ability is a mana ability. Mana abilities don't use the stack and just resolve immediately.

Yes, you can. The type/color of mana a land "could produce" is any type/color of mana any of its mana abilities would generate were they to resolve at that time. You don't have to be able to actually activate those mana abilities. For example, you could tap Reflecting Pool for colorless mana or mana of any color if you controlled only it and Cascading Cataracts, even though there is no way you could activate the "5, T: Add five mana in any combination of colors to your mana pool." ability with your current resources.

City of Brass and Sorrow’s Path, on the other hand, both have triggered abilities that trigger when those lands become tapped for any reason. This means that thoses abilities will trigger even if you use the black mana ability granted by the swamp type.