MTG Salvation

Yes. Turning a permanent face up using morph is a special action (C.R. 702.36e, 115.1), not casting a spell or activating an ability (C.R. 702.60a), so it's not prevented by split second. (Note that casting a card face down is prevented while the spell with split second is on the stack.) And the ability that triggers from turning Willbender face up is not prevented either.

EDIT (Sep. 16): Used a better rule citation.

Becoming the target of a spell or ability is a triggered ability that exists independently of that spell or ability, so even if that spell or ability leaves the stack (see also C.R. 112.7a). In the case of Silverfur Partisan, you will put as many tokens on the battlefield as the triggered abilities that resolve, even if the spells that made them trigger leave the stack.

EDIT (Jun. 6, 2018): Edited in view of rule changes with Dominaria.

An ability that triggers when an object "becomes the target" of a spell or ability triggers while the spell is being cast or the ability activated (C.R. 601.2c) or when the target is changed to that object when it wasn't immediately before (for example, with Redirect).

EDIT (May 9, 2020): Correctness edit.

For the purpose of choosing your commander, an ability of the form "As long as your devotion ... is less ..., [this object] isn't a creature" (such as found in Keranos) is not relevant; it's only relevant while the object with that ability is on the battlefield (C.R. 113.6). It has no effect while it's anywhere else, including in the command zone. Your devotion to a color is the number of mana symbols of that color in the mana costs of permanents you control (C.R. 700.5).

EDIT (Mar. 12, 2017): Minor edits.
EDIT (Apr. 10, 2017; Sep. 11, 2018): Correctness edits.
EDIT (Sep. 3, 2019): Correctness edit. One rule was renumbered with Core Set 2020.
EDIT (Apr. 6, 2020): Clarification.

A permanent that has transformed hasn't entered the battlefield and doesn't become a new game object (C.R. 711.14). Thus, in general, you can attack with a transformed creature as long as you controlled that permanent (whether in transformed or untransformed form) continuously since the beginning of your most recent turn (C.R. 302.6). But remember, transforming a permanent doesn't untap it (C.R. 701.28a), so in this scenario, Grisly Anglerfish will remain tapped, rendering it generally unable to attack during your combat phase (not: combat step) (C.R. 508.1a).

EDIT (Aug. 1, 2019; Sep. 1, 2019): Clarification.
EDIT (Mar. 16, 2020): One rule was renumbered with Core Set 2020.
EDIT (Nov. 26, 2020): Some rules were renumbered in the meantime.

The Eldritch Moon release notes are now available. According to C.R. 712.4c, if an effect tracks a melded permanent and "causes actions to be taken upon" cards that were once that permanent, "the same actions are taken upon each of them." As a result, it would result in an Aura being attached to two permanents, which is not allowed, so in that case, the Aura's controller chooses which permanent to attach it to (C.R. 303.4d, see also C.R. 400.6). Rezzahan turned out to be correct.

EDIT (Apr. 18): C.R. 712.4c, as it appears in the actual comprehensive rules document, has slightly different wording from the same rule as it appears in the Eldritch Moon release notes.

A spell doesn't "targe[t] only Mirrorwing Dragon" if it targets anything other than Mirrorwing Dragon (C.R. 115.9c). Therefore, if it targets a spell or a player in addition to Mirrorwing Dragon, Mirrorwing Dragon's ability won't trigger.

And even if it were to trigger and the spell were copied, the ability doesn't allow the modes of the spell's copies to be changed (C.R. 700.2f-g, 706.10).

EDIT: Note also that if Mirrorwing Dragon's ability triggers, the spell gets to be copied as appropriate before the original spell resolves, not after (C.R. 706.10, 405.2, 116.4).
EDIT (Oct. 27): Added rules citations.
EDIT (Mar. 10, 2019): One rule was renumbered with Dominaria.
EDIT (Jun. 19, 2019): Edit first edit note.
EDIT (Aug. 29, 2019): One rule was renumbered with Core Set 2020.

What is now C.R. 115.6 clarifies that a spell with no targets chosen for it (even if it "require[s] targets", as far as the rules are concerned, by using terms such as "up to one target...", "X target..." and the like in its text) is not a targeted spell; for that reason, such a spell will not fail to resolve merely because it had no targets at all as the spell was cast (C.R. 608.2b applies only if a spell "specifies targets").

Mindclaw Shaman's ability doesn't limit the instant or sorcery cards that can be cast; for that it would have to say "... but only if one or more targets are chosen for it" or the like.

EDIT (Aug. 9, 2017): Correctness edit on first paragraph.
EDIT (Apr. 29, 2018): Edited to conform to changes in Dominaria.
EDIT (Jan. 31, 2019): Edited to conform to changes in Ravnica Allegiance.
EDIT (Oct. 11, 2019): Edited to conform to changes in Core Set 2020.

You're correct, no. Abilities that trigger whenever a spell is cast trigger while that spell is on the stack. Because that triggered ability will go on top of that spell (C.R. 405.2), it will get to resolve before that spell (C.R. 117.4). And Havengul Lich's delayed triggered ability will refer to activated abilities on the card on the stack, where Necrotic Ooze won't have any additional activated abilities.

EDIT (July 29, 2016): Edited for more rules-technical correctness.
EDIT (Mar. 20, 2017): Added rule citations.
EDIT (Jan. 31, 2022): One rule was renumbered in the meantime.

In general, the card discarded with madness won't remain in exile after the turn has ended. When the madness triggered ability resolves, in general, either the card will be cast or it will be put into the graveyard. (Earlier versions of madness used a more complex mechanic for allowing the card to be cast.)

At the time the rules for madness were introduced, there were hardly any ways to put exiled cards (then called "cards removed from the game") into the graveyard, hand, or any other zone, except if the same card exiled them previously. This, perhaps, explains the statement that discarding a card with madness this way keeps it "safe from your opponent’s tricks". (Note that a card in exile can't be "destroyed" [C.R. 701.7a].) In contrast, there were many ways to put cards in a graveyard into another zone, thus making it unavailable for being cast.

You should also know that back then, discarding a card with madness into exile (then called the "removed-from-the-game zone") was optional. It was only very recently made mandatory with Shadows over Innistrad.

EDIT (Jan. 10, 2018): Add rule citation.

Yes. A creature spell is considered a creature spell no matter where it was cast, including from the graveyard (according to C.R. 601.2a, a player moves a card "from where it is to the stack" as the first step of casting it). Somberwald Sage, etc. would have to say "Spend this mana only to cast creature spells from your hand" or the like if the intent is to restrict where the spell can be cast using that mana (an example is Altar of the Lost).

chiamp:

The activated ability uses the phrase "an exiled card named ..." (C.R. 108.1), so it refers to an exiled card with the name Sengir Nosferatu, not to an exiled card named Quicksilver Elemental, etc. (C.R. 201.4b; compare with Mirror-Mad Phantasm). Accordingly, nothing happens if no card named Sengir Nosferatu is in exile when the Bat token's triggered ability resolves (C.R. 609.3, 607.1a).

EDIT (Jan. 3, 2018): Correctness edit.
EDIT (Jun. 12, 2018): Correctness edit to conform to Oracle text for Sengir Nosferatu.
EDIT (Oct. 11, 2018): Correctness edit; add rule citation.
EDIT (Apr. 8, 2019): Add rule citation.
EDIT (May 9, 2019; Jan. 31, 2022): Correctness edit.

cantspeak:

I should repeat: "In my opinion, a question that asks why a particular card is worded a particular way (when a better wording is alleged to exist) is not really suitable for this forum." That's because questions like these (such as those involving the wordings of Confusion in the Ranks and Omniscience) cannot really be answered by analyzing the game's rules, but are often answerable only by the research and development team.

The play is correct. When Cloudshift resolves, it exiles and returns Archaeomancer as a new creature (which isn't blocking anything anymore) (C.R. 400.7, 506.4), triggering Archaeomancer. As Cloudshift finishes resolving, it's placed in the graveyard in time to be targeted by Archaeomancer's ability (C.R. 608.2k, 603.3). Your opponent can then proceed this way with the remaining two Archaeomancers. (While the stack will be empty after each iteration, that alone won't end the declare blockers step [C.R. 500.2].) Meanwhile, your attackers will remain blocked and won't assign combat damage as long as no creatures are blocking them (C.R. 509.1h, 510.1c).

(You probably meant the opponent casts Cloudshift "after" blocking, not "in response to" blocking. To keep your attackers from assigning combat damage, Cloudshift needs to exile and return creatures that are blocking, not those that are about to block.)

You resolve each chosen mode of the spell one after the other, in the order given on the spell (C.R. 608.2c). Thus, you return Ajani's Pridemate to the battlefield in time to see you gain 4 life and make its ability trigger. However, the ability will trigger only once, not for each 1 life you gain (C.R. 118.9) (thus, you put only one +1/+1 counter on Ajani's Pridemate when it resolves).

EDIT (Jan. 22, 2019): Edited to conform to Oracle text change with Ravnica Allegiance.