In a 3 player game how many runes does player 2 channel?

Player 2 channels 2 runes — the standard Channel Phase amount. The "extra rune" bonus in FFA3 (Skirmish) goes only to the player going last (Player 3), not to the middle player (Rule 487.7).

Player 2 channels on their own first Channel Phase, same as any normal turn (Rule 315.3.b / 430.4.a). They also draw a card on their first turn — only Player 1 skips that draw, and only Player 3 gets the extra channel.