The penalty on secondary/iterative attacks is actually a penalty to your Base Attack Bonus, which is used in the Combat Maneuver Bonus calculation. Decreasing your effective BAB for the maneuver reduces your CMB.

The character is definitely “Armed” (which has a technical definition about threatening attacks of opportunities, which natural weapons can do); I definitely would not apply the −4 penalty. On the other hand, clearly your hand is not full and should be able to grab a weapon. It seems to me that those rules were written without consideration for natural weapons or Improved Unarmed Strike, and the re-use of the term “armed” leads to some ambiguity.

Nonetheless, I think it’s pretty fair to allow no −4 penalty but allow weapon-grabbing.

None of these

Normally, an Eidolon only has the attacks provided by its evolutions, so the quadruped form has a bite attack in its stats block because it starts with the bite evolution for free, and has no other attacks. That bite, due being its only attack, will already add 1-1/2 times its Strength modifier on damage rolls, as noted in the same page.

If you later purchase other form of attack (claws evolution, as a example), or gains somehow another natural attack (as you asked us to assume), the bite attack would add only 1x Strength modifier from that point onwards, since it is not anymore its only form of attack. This is when the part you quoted from the bite evolution enters play. Purchasing the bite evolution again will allow your Eidolon to apply the increased modifier always, even if its bite attack is not its only form of attack. But the free bite evolution represents the attack with what the Eidolon was created, if it gains more attacks, it would be later.

As a less serious note, if you go by option 1, then you quadruped eidolon, having 4 legs due being naturally quadruped plus 4 more legs due the free limbs evolutions, would start with a total of 8 legs, and it would not be a Quadruped anymore.