In the situation you've proposed (the first one), Lifeline's triggers will keep returning the Avacyns to the battlefield and they will promptly die from the Legend Rule, once a turn (at the beginning of the end step). And yes, if there are an odd number, one will stay on the battlefield. Whose it is depends on how the triggers get stacked (Lifeline's controller controls all of them, and will thusly get to "choose.").
As for the second question, yes. Any non-combat damage you control that would be dealt to an opponent can be redirected a planeswalker that player controls. This includes mass-targeted or upkeep-triggered sources.
A pertinent clarification is that "cast without paying its mana cost" is, in the above rules quotes, an alternative cost rather than a cost reduction. Some people misinterpret these rules as "without paying its mana cost" meaning "a reduction of all mana." This is not the case, and as such, the additional cost is still applied, and there is no reduction to apply in this scenario.
117.10. Each payment of a cost applies to only one spell, ability, or effect. For example, a player can't sacrifice just one creature to activate the activated abilities of two permanents that each require sacrificing a creature as a cost. Also, the resolution of a spell or ability doesn't pay another spell or ability's cost, even if part of its effect is doing the same thing the other cost asks for.
No. Creatures (and permanents in general) that are sacrificed to pay one cost cannot satisfy the cost of a separate ability. The metaphor that most people typically use is that of a vending machine: You only have one dollar bill, that doesn't allow you to press two buttons on the machine, even though they each just cost one dollar.
You can see this on some older cards, where weird phrases like "you cannot sacrifice cards that are already on their way to the graveyard" explicitly spelled out. Now it's all just implicit in the rules.
(there's probably a rules quote, I'll edit it in or someone else will post it)
Cascade doesn't apply a cost reduction, it provides an Alternate Casting Cost (of no mana). Additional costs (such as "sacrifice a creature") MUST be paid, and optional additional costs (like kicker or multikicker) CAN be paid, but have no "cost reduction" applied.
This is not the case. With cards like Izzet Charm, you must choose your "Mode" as a part of the initial casting of the spell. This is locked in to the spell once it's on the stack, and you can't change it later. This is important for things like copying spells (a copied spell has the same modes), and as you show in your example, knowing how to respond to one. Your opponent gets to know what mode you chose for charm before he or she responds.
From the comp rules:
(on how to cast a spell):
601.2b If the spell is modal the player announces the mode choice (see rule 700.2).
(on modal things in general):
700.2a The controller of a modal spell or activated ability chooses the mode(s) as part of casting that spell or activating that ability. If one of the modes would be illegal (due to an inability to choose legal targets, for example), that mode can't be chosen. (See rule 601.2b.)
Correct and correct. A miracle is cast during the draw step, so Chandra can't be activated in time, but an instant like Increasing Vengeance (or Reverberate) can.
If you only have one precursor golem and the two token golems, a single kicked Rite (targeting a token golem) will leave you with 27 golems, 6 of which are Precursors.
If you do it AGAIN, you get 4 Million. And that's that.
To clarify: Havengul Lich does not remove timing restrictions. Some abilities instruct you to cast spells as a part of their resolution, which /does/ remove a timing restriction, but the Lich simply sets up an effect that lets you cast a card from an unusual zone. It says nothing about timing, so as Phoenix said, you have to wait for when you could normally cast that creature (in the case of most creatures, during your Main Phase when nothing else is on the stack and you have priority).
As for the second question, yes. Any non-combat damage you control that would be dealt to an opponent can be redirected a planeswalker that player controls. This includes mass-targeted or upkeep-triggered sources.
Here's the comp rules as well.
You can see this on some older cards, where weird phrases like "you cannot sacrifice cards that are already on their way to the graveyard" explicitly spelled out. Now it's all just implicit in the rules.
(there's probably a rules quote, I'll edit it in or someone else will post it)
(The only exceptions being cards like Firespout from Shadowmoor that care which mana was used to cast it. Those will specify in the card text.)
From the comp rules:
(on how to cast a spell):
601.2b If the spell is modal the player announces the mode choice (see rule 700.2).
(on modal things in general):
700.2a The controller of a modal spell or activated ability chooses the mode(s) as part of casting that spell or activating that ability. If one of the modes would be illegal (due to an inability to choose legal targets, for example), that mode can't be chosen. (See rule 601.2b.)
Hence the flavor text! Woop woop.
The flavor text that I LOVE, by the way. This card is neat.
@damagecase: White, by virtue of the color pie, gets better removal than green. Also, Path to Exile is undercosted.
If you do it AGAIN, you get 4 Million. And that's that.
If a source you control ever deals damage to a player, you have the choice of redirecting that damage to a planeswalker that player controls.