1. Yes and No. TWF penalties only apply when you are actively using both weapons to attack. "If you wield a second weapon in your off hand, you can get one extra attack per round with that weapon. You suffer a -6 penalty with your regular attack or attacks with your primary hand and a -10 penalty to the attack with your off hand when you fight this way." "This way" refers to getting one extra attack per round. If it did not refer to that, then the only way to avoid taking TWF penalties would be to wield a weapon in two hands. You can make a Shield Bash if you're using a shield, or an unarmed strike otherwise. If you are only holding two weapons, you don't take TWF penalties. If you are using both weapons to attack, and thus gain the extra attack, then you do take penalties. Also, compare the wording of TWF to, say, Snap Kick, which says "You take a -2 penalty on all attack rolls you make this round."
2. Yes. AoOs only care if you threaten the relevant square. Your armor spikes allow you to threaten the squares adjacent to you. Ask him where in the rules he's getting the idea that you can only use one weapon per round.