Can opponents play reaction spells in response to triggered or activated abilities from permanents like Teemo's 'When you play me' ability, Darius's triggered ability, or Vi's activated ability?

Ruling: Yes, opponents can play reaction spells in response to triggered and activated abilities from permanents. While you cannot react to a permanent being played, you can always react to the abilities those permanents trigger or activate.

Sequence:

• A permanent (unit or gear) is played and creates a chain
• Due to rule 538, no player can react to the permanent itself entering the field
• The permanent resolves and enters play
• Any triggered abilities (like "When you play me") or activated abilities then go on the chain
• Players can now play reactions to these abilities since abilities are not permanents

Nuances:

• Triggered abilities that trigger "When you play me" happen after the permanent has been played, creating a new chain that can be reacted to
• Abilities are different game objects from permanents - you can't react to permanents but you can always react to their consequences (abilities/combat)
• In the Teemo example, opponents can play Gust in response to the triggered ability before Teemo gets +4M
• In the Darius example, opponents can play Shakedown before Darius gets +2M
• In the Vi example, opponents can play Shakedown in response to Vi's activated ability