#activated abilities

When you tap creatures to activate a Crew ability, you're tapping them to crew, not for their own ability. You're not activating those abilities and they won't happen. Tapping is a cost of both Crew and their own ability, so you can't pay both of those costs with the same creature on the same turn.

For example, you can either tap your Elvish Mystic for its own ability to add G or to crew Brute Suit, but not both with the same tap.

While in Gamer Trance, you cannot perceive enemy intents or the passage of time.

While in Gamer Trance, you do not suffer negative effects from physical debuffs such as fatigue, hunger, discomfort, or wounds. Negative effects still accumulate and are applied cumulatively when you exit Gamer Trance. You cannot perceive the presence or severity of these debuffs until you exit Gamer Trance.

Before.

Discarding a card is part of the cost of Yawgmoth’s proliferate ability, so Archfiend of Ifnir will trigger when you pay that cost and its trigger will go on the stack above the Proliferate ability. 

Yes, this means that all your opponents’ creatures will have -1/-1 counters on them for you to proliferate.

No, it won’t.

Rowan’s emblem will copy activated abilities (other than mana abilities), but it doesn’t have any interaction with triggered abilities such as ETB triggers.

An activated ability is always formatted as “[COST]: [EFFECT]” and is something you choose to have happen by paying a cost. For example, a planeswalker’s loyalty ability is an activated ability and Rowan’s emblem would copy that ability when you activate it. (You don’t pay double the cost though.)

No.

Gaining life from an activated or triggered ability of an Instant or Sorcery card, or from a delayed triggered ability created by an instant or sorcery spell, isn’t the same as the instant or sorcery spell causing you to gain life. The activated or triggered ability is causing you to gain the life, and so Firesong and Sunspeaker won’t trigger.

This is why the much touted “combo” of Firesong and Sunspeaker with Spiritualize doesn’t actually go infinite. Or do anything other than cause you to gain a bit of extra life, really.

This can’t happen the way you describe. By the time you’ve activated the ability, Indulgent Aristocrat is already in your graveyard. It’s not on the battlefield so your opponent can’t target it with a removal spell anymore.

Even if you sacrifice another creature to Indulgent Aristocrat, your opponent removing it in response to the activation won’t stop the ability from happening. Once the ability has been activated, it will resolve regardless of what happens to the source of the ability.

You're going to get zapped.

Whenever something triggers from a spell being cast or an ability being activated, that triggered ability goes on the stack above the spell or ability that triggered it and resolves first. So when Gideon goes to put that bubble around Thunderbreak Regent, the Regent gets mad and gets off a shot at you before Gideon can work his magic.

Sure thing!

When you sacrifice Yavimaya Elder to activate its “2, Sacrifice Yavimaya Elder: Draw a card.” ability, its triggered ability triggers. The trigger goes on the stack above of the activated ability and resolves first. Only after you’ve searched your library for basic lands and reshuffled will the activated ability be able to resolve.

TL;DR: Search first, then draw.