Triggered abilities, such as Butcher of Malakir's, don't go onto the stack until the next time a player would receive priority. Before that, the currently resolving spell or ability has to finish resolving.
The effects of Sheer Drop with the awaken cost paid are to destroy the creature, then put +1/+1 counters on a land and make it a creature. All of this happens before anyone gets priority; therefore, the land is a creature by the time Butcher of Malakir's ability goes onto the stack - let alone resolves. If your opponent's only creature is that land, they'll have to sacrifice it.
is it still a relentless dead when it hits the graveyard
To be precise: Mirage Mirror isn't still a copy of Relentless Dead while it's in the graveyard... but that doesn't matter. Abilities that trigger when something leaves the battlefield* look at the object that was on the battlefield, not the one it became in the new zone. Since the Mirror was a Relentless Dead at the point just before it died, you'll get the triggers.
*One notable disclaimer: There are things that look like a leaves-the-battlefield trigger but aren't, such as Emrakul, the Aeons Torn's "is put into a graveyard from anywhere" trigger. Since this one doesn't specify that it has to come from the battlefield, it isn't a leaves-the-battlefield trigger.
Once an ability is activated, it exists independently from its source. This principle usually applies to removal of the source, but it also applies to text-changing effects - essentially, the text of the ability while it's on the stack is the text that the ability had when it was activated. The emblem would still apply to Swamps unless you applied the text-changing effect to Liliana before it was activated.
You are correct. Master Biomancer's ability applies a replacement effect to creatures entering the battlefield. This means that the Impersonator is never on the battlefield without the +1/+1 counters granted by Biomancer's ability.
Compare with triggered abilities (e.g. Cathars' Crusade or the Graft ability) which happen after the creature has entered the battlefield; this would be too late for an Impersonator that didn't copy something.
(If the intent was to have the mana pay for Guardian's ability, that isn't going to actually work... The mana would be generated during the Declare Attackers step and would be removed from the pool by the time combat damage happens.)
Ghitu Warlord1RR
Creature - Human Warrior
Battle cry R: Ghitu Warlord gets +1/+0 until end of turn.
2/2
Forgotten Lineage1BB
Instant
Name a creature type. Target opponent reveals his or hand and discards all cards of that type. If no cards are discarded this way, draw a card.
Templating for that seems to be "Whenever a permanent is turned face up" according to similar cards on Gatherer.
Enraged Scorpion1B
Creature - Scorpion
Morph B
Deathtouch
If a creature an opponent controls would fight another creature you control, it fights Enraged Scorpion instead.
1/1
Glory of Combat1RW
Enchantment - Aura
Enchant creature
When Glory of Combat enters the battlefield, put a +1/+1 counter on each creature you control.
Whenever you attack with one or more creatures, put a +1/+1 counter on enchanted creature.
Aether Blockage1U
Instant
Exile target spell. At the beginning of the next upkeep, the owner of the exiled card may cast it without paying its mana cost.
Unburdened Insight1WU
Instant
Sacrifice any number of noncreature, nonland permanents. You gain X life and draw X cards, where X is the number of permanents sacrificed this way.
quotation from the first paragraph: "You can, however, pay additional costs."
This means that when you cast a card without paying its mana cost, you can still pay additional costs (e.g. optional ones like kicker) when casting it. For example, if you cast Rite of Replication for free via As Foretold, you can still pay 5 to kick it. You don't get the kicker for free, but neither are you banned from using it (this is different from alternate costs, where you can only choose one alternate cost and As Foretold's 0 is one.)
quotation from the second paragraph: "Ignore any alternative costs, additional costs, cost increases, or cost reductions that could apply to it."
This ruling specifically refers to how you determine an object's converted mana cost. Additional costs don't factor in. Using Rite of Replication as an example, its CMC is 4 under all circumstances. It doesn't become 9 when you kick it.
Your example isn't really a question of over-assigning damage, but yes, you must still assign at least 3 damage to the 0/3 creature before you can assign any damage to the 0/2 creature. You can assign just the 3 damage, or you can assign as much as the full 5 damage, to the 0/3. (The latter could be useful if, for example, the prevention effect is only preventing the first 2 damage and you really want the 0/3 to die - or if the 0/2 has an ability that would trigger by being dealt damage or dying and you don't want that to trigger.)
Thank you for the answer. What are some of other abilities like devotion that aren't affected by OS/Humility?
Basically anything that applies a continuous effect before the ability-modifying layer (and some in the same layer, depending on timestamps/dependency - although this is rare because "loses abilities" tends to create a dependency with other effects that would grant abilities, typically on a conditional basis, thus making Humility win out.) The other layers are, in order:
Copy and control-changing effects usually aren't handled via static abilities in a way that interacts here (Control Magic and similar would be an example, but I can't think of any ability-removing effects that can hit Auras) and text-changing effects are typically created by one-shot spells and similar. Type-changing effects are easier to give examples of; for instance, Duplicant would retain the creature types it gains from the exiled card and Adaptive Automaton would retain the creature type you chose for it even if an effect removes those abilities later. Painter's Servant still makes everything the chosen color even if its ability is removed.
This is a layers situation. There are several effects that apply here, in layer order:
Base characteristics: Heliod is a 5/6 Legendary Enchantment Creature - God with the printed abilities.
Layer 4: If devotion is less than 5, Heliod isn't a creature.
Layer 6: Creatures lose abilities.
Layer 7b: Creatures are 1/1.
Heliod essentially has two possible states. With devotion less than 5, it's not a creature (per its ability that has the effect in layer 4) and therefore the ability of Overwhelming Splendor doesn't apply to it in layers 6 and 7b. With devotion at 5 or more, Heliod's "I'm not a creature" ability doesn't apply, and Splendor proceeds to apply its effects that cause Heliod to lose its abilities and be 1/1.
Having the ability removed by Overwhelming Splendor does not cause Heliod to become unable to stop being a creature. Ability-removing effects are weird in that way. When you apply continuous effects, you always start with the base characteristics and go layer by layer to apply them, which essentially means that any effect from an ability that applies before layer 6 is still able to function even if that ability is being removed by another effect.
The effects of Sheer Drop with the awaken cost paid are to destroy the creature, then put +1/+1 counters on a land and make it a creature. All of this happens before anyone gets priority; therefore, the land is a creature by the time Butcher of Malakir's ability goes onto the stack - let alone resolves. If your opponent's only creature is that land, they'll have to sacrifice it.
To be precise: Mirage Mirror isn't still a copy of Relentless Dead while it's in the graveyard... but that doesn't matter. Abilities that trigger when something leaves the battlefield* look at the object that was on the battlefield, not the one it became in the new zone. Since the Mirror was a Relentless Dead at the point just before it died, you'll get the triggers.
*One notable disclaimer: There are things that look like a leaves-the-battlefield trigger but aren't, such as Emrakul, the Aeons Torn's "is put into a graveyard from anywhere" trigger. Since this one doesn't specify that it has to come from the battlefield, it isn't a leaves-the-battlefield trigger.
Compare with triggered abilities (e.g. Cathars' Crusade or the Graft ability) which happen after the creature has entered the battlefield; this would be too late for an Impersonator that didn't copy something.
Ghitu Warlord 1RR
Creature - Human Warrior
Battle cry
R: Ghitu Warlord gets +1/+0 until end of turn.
2/2
Instant
Name a creature type. Target opponent reveals his or hand and discards all cards of that type. If no cards are discarded this way, draw a card.
Enraged Scorpion 1B
Creature - Scorpion
Morph B
Deathtouch
If a creature an opponent controls would fight another creature you control, it fights Enraged Scorpion instead.
1/1
Enchantment - Aura
Enchant creature
When Glory of Combat enters the battlefield, put a +1/+1 counter on each creature you control.
Whenever you attack with one or more creatures, put a +1/+1 counter on enchanted creature.
Instant
Exile target spell. At the beginning of the next upkeep, the owner of the exiled card may cast it without paying its mana cost.
Instant
Sacrifice any number of noncreature, nonland permanents. You gain X life and draw X cards, where X is the number of permanents sacrificed this way.
Instant
Buyback - Sacrifice a creature.
Target creature you control gets +1/+1 and gains indestructible until end of turn.
This ruling specifically refers to how you determine an object's converted mana cost. Additional costs don't factor in. Using Rite of Replication as an example, its CMC is 4 under all circumstances. It doesn't become 9 when you kick it.
Layer 1: Copy effects
Layer 2: Control-changing effects
Layer 3: Text-changing effects
Layer 4: Type-changing effects
Layer 5: Color-changing effects
Copy and control-changing effects usually aren't handled via static abilities in a way that interacts here (Control Magic and similar would be an example, but I can't think of any ability-removing effects that can hit Auras) and text-changing effects are typically created by one-shot spells and similar. Type-changing effects are easier to give examples of; for instance, Duplicant would retain the creature types it gains from the exiled card and Adaptive Automaton would retain the creature type you chose for it even if an effect removes those abilities later. Painter's Servant still makes everything the chosen color even if its ability is removed.
Base characteristics: Heliod is a 5/6 Legendary Enchantment Creature - God with the printed abilities.
Layer 4: If devotion is less than 5, Heliod isn't a creature.
Layer 6: Creatures lose abilities.
Layer 7b: Creatures are 1/1.
Heliod essentially has two possible states. With devotion less than 5, it's not a creature (per its ability that has the effect in layer 4) and therefore the ability of Overwhelming Splendor doesn't apply to it in layers 6 and 7b. With devotion at 5 or more, Heliod's "I'm not a creature" ability doesn't apply, and Splendor proceeds to apply its effects that cause Heliod to lose its abilities and be 1/1.
Having the ability removed by Overwhelming Splendor does not cause Heliod to become unable to stop being a creature. Ability-removing effects are weird in that way. When you apply continuous effects, you always start with the base characteristics and go layer by layer to apply them, which essentially means that any effect from an ability that applies before layer 6 is still able to function even if that ability is being removed by another effect.