To quote the 3.5 rulebook:
Because a whip can wrap around an enemy's leg or other limb, you can make a trip attack with it. If you are tripped during your own trip attempt, you can drop the whip to avoid being tripped.
When using a whip, you get a +2 bonus on opposed attack rolls made to disarm an opponent (including the roll to keep from being disarmed if the attack fails).
A 15 foot flank trip attack, where the bad-guy getting up provokes an attack of opportunity from the main fighter is incredibly powerful, even if you do no "damage".
These same rules are in Pathfinder (pgs 143-145). Whips are listed as disarm, nonlethal, trip weapons on the table on page on 143, then from 144-145:
Disarm: When you use a disarm weapon, you get a +2
bonus on Combat Maneuver Checks to disarm an enemy.
Nonlethal: These weapons deal nonlethal damage (see
Chapter 8).
Trip: You can use a trip weapon to make trip attacks. If
you are tripped during your own trip attempt, you can
drop the weapon to avoid being tripped.
Again ... that trip attack is invaluable. The only "more broken" weapon is the spiked chain in 3.5, which does threaten the 10' reach and allows a trip attack, which means you get a free trip attack on bad-guy again when he tries to get up, along with your normal attacks once you trip him (one of my favorite characters ever was a Dwarf Fighter specializing in a spiked chain).
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).
Best Answer
As written, yes, you can combine that with a reach weapon.
3.X and PF rules are descriptive; that is, they generally tend to tell you what you can do. The RAW on the statement indicates that you can pick one attack - in the scenario you described, that'd be either 'manufactured weapons' or a specific weapon depending on the DM's interpretation of "one" - and the Eidolon gains another five feet of reach with it. Gaining reach in this fashion is fairly standardized, though this particular ability is not, and the flavor is admittedly a little odd.