claws are all primary attacks, so they have no penalty. Primary attacks are at the full bonus, secondary attacks are at -5 or -2 with the multi attack feat.
That said, what level is the summoner? He must be at least 4th level to be able to have 4 natural attacks (before that the maximum is 3).
I was wondering how his attacks would be calculated. Does he have four claw attacks at his highest base attacks bonus and adds his strength modifier or does he have any penalties.
Yep, full base attack bonus, and full strength bonus on each attack. No penalties for full attacking with multiple natural attacks, as long as they are primary attacks.
Don't forget that those claws are both slashing and bludgeoning for the purpose of overcoming damage reduction.
Bites are slashing, bludgeoning, and piercing.
And to make a note on the 4 nat attacks, even if the summon can only make 3 attacks, he can still have all four arms with attacks available - he can just only make 3 attacks until the summoner is high enough level.
And that's only on a full-attack round. Without a full attack, he only gets one. And don't forget that the rules state that you make your first attack, and then you can choose to continue with a full attack or do something else (like move more than 5 feet, or pull out an item). You can also make an attack, see the effects, and then direct your other attacks to another monster. You can even make an attack, take a 5 foot step, and then continue the rest of your full attack. And you don't have to call targets before the attacks.
Rules here (scroll down to "Full Attack":
Full Attack
If you get more than one attack per round because your base attack bonus is high enough (see Base Attack Bonus in Classes), because you fight with two weapons or a double weapon, or for some special reason, you must use a full-round action to get your additional attacks. You do not need to specify the targets of your attacks ahead of time. You can see how the earlier attacks turn out before assigning the later ones.
The only movement you can take during a full attack is a 5-foot step. You may take the step before, after, or between your attacks.
If you get multiple attacks because your base attack bonus is high enough, you must make the attacks in order from highest bonus to lowest. If you are using two weapons, you can strike with either weapon first. If you are using a double weapon, you can strike with either part of the weapon first.
Deciding between an Attack or a Full Attack: After your first attack, you can decide to take a move action instead of making your remaining attacks, depending on how the first attack turns out and assuming you have not already taken a move action this round. If you've already taken a 5-foot step, you can't use your move action to move any distance, but you could still use a different kind of move action.
And to make a note on the 4 nat attacks, even if the summon can only make 3 attacks, he can still have all four arms with attacks available - he can just only make 3 attacks until the summoner is high enough level.
Actually the maximum attacks rule state that you cannot take evolutions that would make you surpass the maximum number of attacks.
Though folks could I suppose argue the 'at its maximum' wording to try to get around the seeming intent.
I'll do so just for completeness.
Max. Attacks: "This indicates the maximum number of natural attacks that the eidolon is allowed to possess at the given level. If the eidolon is at its maximum, it cannot take evolutions that grant additional natural attacks. This does not include attacks made with weapons."
A level 1 summoner with a bipedal eidolon has claws, limbs (arms), and limbs (legs). Two natural attacks. He is not at his maximum number, so he can still take evolutions that grant additional natural attacks. So he takes claws (and either applies it to his legs, or also gets an additional limbs evolution) and ends up with 4 natural attacks, of which he can only make 3.
Alternately, it could be considered that when he takes that second claws evolution, it only gives him one more claw, since he is not allowed to possess more than 3. (Though he would still have one unclawed arm which he could use to make weapon attacks or whatever)
Even more alternately, one could consider taking the second claws evolution illegal since it would put him above the maximum number. Which kind of sucks if he's taking other evolutions or abilities that all modify the same type of natural weapon. (But maybe that's good, given how strong most summoners tend to be)
Oops. Looks like I got that part wrong. :)
It's been a bit since I played a summoner.
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It says 'if the eidolon is at its maximum, it cannot take evolutions that grant additional attacks' Cool. If you're at 2, you are allowed to take evolutions that grant additional attacks. You take claws, this adds 2, now you are at four. By the rules, you can no longer, at this point, take evolutions that grand additional natural attacks.
At no point does it say anything else.