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Our fighter is exhausted and is suffering a -6 to Dexterity as a result, bringing his Dexterity from 15 to 9.

He has the feat Two-Weapon Fighting, which reduces the penalty for fighting with two weapons. The feat has a prerequisite of 15 Dex.

Will exhaustion preclude him from getting the benefit of the Two-Weapon Fighting feat?

No, only Ability Drain does

This is what James Jacobs (Paizo's Creative Director) said about this (original post):

I am confused on the different types of Ability Damage, Drain, and Penalty. Plus some spells like Ray of Enfeeblement and Touch of Idiocy seem to even have special rules. How do these effects really work?

Ability damage only results in a penalty to actions associated with that ability score; it does NOT make you lose access to feats or spells that require ability score minimums, since your actual ability score does not lower. Only ability DRAIN can make you lose access to spells you can cast or feats that have prerequisites.

But it's not that simple. Some effects that cause ability damage or ability penalties DO have additional effects. Touch of idiocy is one such spell, since it says in the spell's description that it affects the target's ability to cast some or all of its spells if the penalty imparted to the ability score drops low enough. This is an exception to the general rule for ability scores and applies only to touch of idiocy (the point of the spell, really, is to be a lesser version of feeblemind that screws over spellcasters, after all).

Ray of enfeeblement, on the other hand, does NOT have this type of language. It merely works as a normal penalty to an ability score.

1. 
   

   A 12th-level wizard with a 16 Intelligence takes a 3 point Intelligence penalty from Touch of Idiocy. What is the highest level spell he can cast, sixth or third?

   
   
   

   Third.

   
   


2. 
   

   A 12th-level wizard with a 16 Intelligence takes 3 points of Intelligence damage. What is the highest level spell he can cast, sixth or third?

   
   
   

   Sixth.

   
   


3. 
   

   A 12th-level wizard with a 16 Intelligence takes 3 points of Intelligence drain. What is the highest level spell he can cast, sixth or third?

   
   
   

   Third.

   
   



4. A fighter with 13 strength and the Power Attack feat takes a 3 point Strength penalty from Ray of Enfeeblement. Can he Power Attack?
   Yes.



5. 
   

   A fighter with 13 strength and the Power Attack feat takes 3 points of Strength Damage. Can he Power Attack?

   
   
   

   Yes.

   
   


6. 
   

   A fighter with 13 strength and the Power Attack feat takes 3 points of Strength Drain. Can he Power Attack?

   
   
   

   No.

   
   

This clarification is consistent with the rules about Ability Drain, which says:

Ability drain actually reduces the relevant ability score. Modify all skills and statistics related to that ability. This might cause you to lose skill points, hit points, and other bonuses. Ability drain can be healed through the use of spells such as restoration.