Interestingly, I'm pretty sure you COULD return them if they got exiled, because the ability doesn't say "return from your graveyard to the battlefield." Colossal Grave-Reaver would still find the card in exile if it ended up there between being milled and the trigger resolving. However, the same isn't true for your library because that's a hidden zone and nothing is letting you search it.

TL;DR: Yes, you can pick the order. Either way, your graveyard will get shuffled, whether it includes an Eldrazi is up to you.

When effects trigger, they aren't put on the stack immediately (Rule 603.2). They wait until the next time players would receive priority (603.3). Essentially, a triggered ability happens in three stages: First, the game state matches the trigger. Second, when players would get priority, all waiting triggers are put onto the stack. Third, they resolve with the usual process.

In this case, "Whenever one or more creatures are put into your graveyard from your library" and "Whenever *this is put into a graveyard from anywhere" both trigger when Colossal Grave-Reaver mills an Eldrazi titan. Once you finish milling, all waiting triggers are put onto the stack in APNAP (active player, nonactive player) order. Since it's your turn, you'll put your triggers on the stack in any order you like, then other players will add theirs in turn order if they triggered any. At this point you can put Colossal Grave-Reaver's trigger on top if you want to stick an Eldrazi on the battlefield (and why wouldn't you?) or an Eldrazi trigger on top if you want them shuffled away.

When the abilities resolve, you end up with one of the following options:

1. You put an Eldrazi trigger on top. Your graveyard gets shuffled. Then Colossal Grave-Reaver's trigger resolves, fails to find any of the creature cards that got milled, and doesn't return anything.
2. You put Colossal Grave-Reaver's trigger on top. It returns one of the milled creature cards to the battlefield. Then the Eldrazi trigger resolves and shuffles your graveyard into your library.

Note: They may also not have a supertype or subtype at all, and that's okay too!

Yes. There's no such thing as a 'dual type'; cards may have one or more supertypes (basic, legendary, snow, etc), one or more card types (land, artifact, creature, etc) and one or more subtypes (Vehicle, Elf, Aura, Vraska, etc). Each of these types can be found by any card looking for them. If you cast Umori, the Collector and choose artifact, any card with Artifact in its type line counts. Rabbit Battery, Sol Ring, Darksteel Citadel, etc are all artifacts.

This is detailed in rule 616.1 (interaction of replacement and/or prevention effects). Since both of these affect the same player, the affected player chooses which order to apply them in. This usually but not always means, in practice, that the defending player will choose to prevent the damage with Nine Lives, though it's notable here that they can choose to prevent the damage with Undead Alchemist if they wish to not get more incarnation counters.

So in general, Nine Lives wins.

In this case, order matters. Xenic Poltergeist and Swift Reconfiguration are both type-changing effects, so the most recent timestamp will apply. First, Xenic Poltergeist will make your Sol Ring into an artifact creature, and then Swift Reconfiguration will make it into a noncreature Artifact - Vehicle.

If you were to activate Xenic Poltergeist again (with a second copy, or on a later turn), that effect would now have the most recent timestamp and your Sol Ring would now be a 1/1 Artifact Creature - Vehicle.

Neotrup that's actually clever. Sagas have small text boxes, I never considered that "after your draw step" might be a templating shortcut for "at the beginning of your precombat main phase."

Shifting Woodland becoming a copy of a card doesn't involve it entering the battlefield. It gains the card's copiable characteristics as soon as the ability resolves. With that in mind:

1. Shifting Woodland will become a Nissa of Shadowed Boughs with no loyalty counters on it (as it didn't enter the battlefield), and immediately after the copy ability resolves it will be put into your graveyard as a state-based action. This happens before you gain priority, so you will not be able to activate any abilities.

2. By default, the rooms on a Room card are locked. When you cast one side of the card, that room unlocks as it enters. Since Shifting Woodland isn't copying a spell, it will become a copy of Walk-In Closet / Forgotten Cellar with both rooms locked. You have to pay the cost to unlock each room you want to use. See the Oracle ruling: If a Room enters from any zone other than the stack, it will enter with both halves locked. (This isn't entering that way, but it's also not entering at all.)

3. Yes, you can level up a Class card that Shifting Woodland becomes a copy of. It won't keep the levels once it stops being a copy, but it does gain all the abilities of Scavenger's Talent including the ability to level up for the turn.

4. No abilities. See the Oracle ruling: If you somehow control Barrensteppe Siege and no choice was made for it (perhaps because another permanent on the battlefield became a copy of it), it has neither of the two abilities.

5. You don't get priority during the untap step, but you do gain priority during your upkeep. At this point, you can activate Shifting Woodland and make it a copy of a Saga like Battle for Bretagard. After the upkeep step is your draw step, which will trigger the Saga as usual. At the end of the turn (during the cleanup step), Shifting Woodland will stop being a copy, but the lore counter will remain—if you do this again, you will get the II ability of Battle for Bretagard, and if you do it a third time it will trigger the III ability and then sacrifice itself.