#infamousktf

That depends on whether your opponent gives you the opportunity. Any time a spell or ability resolves, the active player (whoever’s turn it is) is the first player to get priority (the chance to do things).

So in your example, after the planeswalker enters the battlefield (assuming it’s your opponent’s turn, which it probably is if they’re casting a planeswalker), your opponent gets priority again. If they choose, they can immediately activate one of the planeswalker’s abilities (again, assuming it’s their main phase and there’s nothing else on the stack). Once they do that, you can cast Hero’s Downfall in response to the ability — but at that point, the ability is already on the stack. Hero’s Downfall will destroy the ‘walker, but then the ability will still resolve.

On the other hand, let’s say your opponent doesn’t immediately activate a planeswalker ability, and instead announces they’re going to move to the next phase of their turn (or cast another spell, or activate an ability of another card). In that situation, you’d get priority in response to that action, so you’d be able to cast that Hero’s Downfall before they used any of their planeswalker’s abilities.

Priority is a very complicated topic (as is the stack in general), so if this doesn’t make 100% sense, feel free to ask any followup questions you need to. I also recommend checking out my older tutorials, specifically:

MTG Tutorials #1: What Is The Stack?

MTG Tutorials #2: All About Activated Abilities

MTG Tutorials #4: What Is Priority?

Nope. Once an emblem is created, it persists for the rest of the game.

EDIT

anerdtumblr said: 

Karn can restart the game removing the emblems :v

Yes, okay. Karn restarts the game in its entirety, so I suppose you can consider him a way to get of emblems. He also resets players’ poison counters, life totals and commander damage.

No and no.

Auras and equipment that are no longer attached to something they can legally enchant/equip immediately “fall off”. For Auras, this means being put into their owner’s graveyard; for Equipment, this means leaving them on the battlefield but unattaching them from the creature they were attached to.

No. It will enter with a number of counters equal to its printed loyalty.

Planeswalkers always enter the battlefield with counters equal to their printed loyalty (barring effects like Doubling Season, of course.) 

-- EDIT -- 

Apologies everyone, I misread infamousktf's question!

The answer to this question is yes, you can target an untapped Mountain with Koth's first ability. The only restriction for "Untap target Mountain" is that it has to target a Mountain. Note that untapping something that's already untapped doesn't do anything, but the land will still be turned into a 4/4 creature.

If the ability said, "Untap target tapped Mountain," you would not be able to legally activate it targeting anything but a tapped Mountain.

Sacrificing Quest for Pure Flame creates a replacement effect which lasts for the rest of the turn. If a source you control would deal damage while that replacement effect is around, instead that source deals twice as much damage.