Can a creature with two natural weapons, say two claws,
make two grapple attempts in a single round? Could a
character using the rules for fighting with two weapons pull
off the same trick? Exactly how many grapple checks can
you make each round?
You can make one grapple check for every attack your base
attack bonus normally allows. The number of weapons you use
doesn’t affect the number of grapple checks you can make.
For example, a dire ape, an animal with 5 Hit Dice, has a base attack bonus of +3, so it can make only one grapple check each round despite its three natural weapons. Note that when you opt to deal damage with a grapple, you deal subdual damage according to your size, not your natural weapon.
For example, a dire ape (a Large creature) deals 1d4 points of subdual damage with a successful grapple attack, plus its Strength bonus of +6.
Note that creatures with the improved grab special attack can use their natural weapons in a grapple. Note also that monk characters deal their unarmed strike damage with successful grapple checks. Monks and creatures with improved grab are formidable grapplers.
The normal grappling rules assume that the attacker uses
pretty much her whole body in the attack. (When you grapple,
you wrap your arms and maybe your legs around the foe.) You
can grapple while using a shield, but the shield's armor check
penalty applies to your grapple check.
If the creature has the improved grab special attack, it can opt to use only one appendage to make a grapple attack, but it suffers a -20 penalty to its grapple checks if it does so. A character also could use the -20 option, but as you point out, the character also would have to use the two-weapon fighting rules. See Table 8-2 in the Player's Handbook, and remember that a character's own hands are considered light weapons.
The two-weapon fighting penalties apply to both grab
attempts and any grapple checks that occur after successful
grabs. The -20 penalty applies only to grapple checks. For
example, a character who has both the Two-Weapon Fighting
and Ambidexterity feats would suffer a -2 penalty to each grab
attempt and a -22 penalty to any grapple check she makes after
a successful grab.
On her turn, the character can stop making separate grapple
checks with each hand and instead use both hands on one
opponent; if the character does so, neither the two-weapon
penalties nor the -20 apply to grapple checks she makes that turn.