If you don't have multiple attacks from your BAB but you have a double weapon like a quarterstaff, do you get multiple attacks if you take a full-round action or do you still need the bonus from the BAB for that to be valid?
Mixing weapon and natural attacks is a mess. Fortunately, it is not what happens in this example.
Here you see 2 full attack options: one for 4-hand multiweapon fighting (iterative attacks from main hand, single attacks at highest BAB from off hands, no penalty for multiweapon fighting due to racial ability) and one for all-natural full attack (several natural attacks at highest bonus each). And it looks like this creature has 6 hands, but only wields weapons in 4 of them.
Yes, the spear ignores negative progression because it is used in different hand.
Ever-helpful Pathfinder SRD link, just in case.
You get one attack roll for each natural attack you possess, and you do not get any extra natural attack for high BAB (you still gain extra weapon attacks for high BAB if you are using a weapon)
In the case of the Oliphant exposed, because both the Gore and Stamp attacks are primary, you get all the three attacks at your full BAB, and no extra attack because of a high BAB. Assuming your size is Huge:
Melee gore +11 (2d8+5), 2 stamps +11 (2d6+5)
Bonus break down: +8 BAB +5 Str -2 size (because all the three attacks are of primary type, the -5 penalty to secondary attacks does not apply)
The confusion happens because the no BAB extra natural attacks is not clearly stated in the monster rules natural attacks entry, but it is specified in the combat chapter in the natural attacks entry under the Attack section as referenced in the question.
For a full explanation about BAB and natural attacks, first of all, the rule which states you get extra attacks as your Base Bonus Attack (BAB) rises is under common terms definition:
Base Attack Bonus (BAB): Each creature has a base attack bonus and it represents its skill in combat. As a character gains levels or Hit Dice, his base attack bonus improves. When a creature's base attack bonus reaches +6, +11, or +16, he receives an additional attack in combat when he takes a full-attack action (which is one type of full-round action—see Combat).
So by the BAB rule you get additional attacks when you reach certain BAB levels.
Let's check what the natural attacks rules say:
Natural Attacks: Most creatures possess one or more natural attacks (attacks made without a weapon). These attacks fall into one of two categories, primary and secondary attacks. Primary attacks are made using the creature's full base attack bonus and add the creature's full Strength bonus on damage rolls. Secondary attacks are made using the creature's base attack bonus –5 and add only 1/2 the creature's Strength bonus on damage rolls. [...]
Creatures with natural attacks and attacks made with weapons can use both as part of a full attack action (although often a creature must forgo one natural attack for each weapon clutched in that limb, be it a claw, tentacle, or slam). Such creatures attack with their weapons normally but treat all of their natural attacks as secondary attacks during that attack, regardless of the attack's original type.
So nowhere says natural attacks do not get extra attacks for high BAB, but it neither says they do, and it makes a clear difference between natural attacks and attacks made with weapons. So the intention seems clear that natural and weapon attacks are different things and work differently.
The confusion is cleared when you check the natural attacks entry of the Combat section:
You do not receive additional natural attacks for a high base attack bonus.
We can also infer this works like that from examples. In this case take a look to the Bear in Paizo Bestiary. At one side we have the Grizzly Bear (BAB +3) at the other the Dire Bear (BAB +7):
Bear, Grizzly: Melee 2 claws +7 (1d6+5 plus grab), bite +7 (1d6+5) Base Atk +3; CMB +9 (+13 grapple); CMD 20 (24 vs. trip)
Bear, Dire: Melee 2 claws +13 (1d6+7 plus grab), bite +13 (1d8+7) Base Atk +7; CMB +15 (+19 grapple); CMD 26 (30 vs. trip)
As you can see the Dire Bear has +7 BAB, which corresponds to an attack iteration of +7/+2, but if you look at the Dire Bear attacks, he has exactly the same number and type of attacks than Grizzly bears. Thus, it seems that Pathfinder developers intention is high BAB does not give you access to additional natural attacks, and that is something reserved to weapon attacks, unnarmed strikes and combat maneveurs.
Remember you need to take a full-attack action in order to perform all your natural attacks, and you still benefit from additional attacks if you are holding at least one weapon.
Addendum about natural attack bonuses
When performing multiple natural attacks, you have also to take into account the difference between primary and secondary attacks. So let's see when a natural attack is considered primary or secundary based on the universal monster rules:
Primary: You attack with full BAB and full Strength damage bonus. Are considered primary:
- Typical primary natural attacks: Bite, Claw, Gore, Slam, Sting, Talons
- Whatever attack if it is the only type of natural attack the creature has (be it single attack as a Tile Slap, or multiple as 2 Pincers)
Secondary: You attack with BAB -5 and half Strengh damage bonus. The multiattack feat reduces this penalty from -5 to -2. Are considered secondary:
- Typical secondary natural attacks: Hoof, Tentacle, Wing, Pincers, Tail Slap, Others...
- Whatever attack if it is used in combination with weapon attacks (including unarmed strikes).