1)If a creature take -20 to CMB for grapple, have the -20 at CMD, too?
Grab (Ex): "The creature has the option to conduct the grapple normally, or simply use the part of its body it used in the grab to hold the opponent. If it chooses to do the latter, it takes a –20 penalty on its CMB check to make and maintain the grapple, but does not gain the grappled condition itself."
The penalty is only to make and maintain the grapple, it does not apply to anything else.
2) Spellbook of beast within. If an alchemist take the greater mutagen, what happens?
Journal of the Beast Within - Beastly Concoction (Su): "This journal grants you the ability to make an extremely potent and savage mutagen. Spend this boon when you imbibe your mutagen. The mutagen grants you a +6 alchemical bonus to your selected ability score instead of the normal +4; you take a –2 penalty to all of your mental ability scores (Intelligence, Wisdom, and Charisma) while the mutagen is in effect."
If you have Greater Mutagen, then you probably shouldn't use the Beastly Concoction, since it's not as good.
3) Can an inquistor with the anger inquisition take the raging vitality?
Raging Vitality: "Prerequisites: Con 15, rage class feature."
Divine Anger (Ex): "At 6th level, you gain the ability to rage like a barbarian."
RAW, no, Divine Anger is not the rage class feature. It seems like a very reasonable house rule, though.
4) Can blightening critical applied to spellstrike?
Blighted Critical: "Whenever you confirm a critical hit with a touch spell, ranged touch spell, or spell-like ability against an opponent, the victim gains a random minor spellblight (see page 95)."
I think so. You're still critting with a touch spell, just not making a touch attack. Same as if the caster was using unarmed strike or a natural attack, it's still a critical hit with the touch spell, regardless of how it was discharged.
5) When the action of spend ki pool/arcane pool is not write, is a swift action?
Examples?
1. No.
2. I don't understand the question.
3. Using the rules as Written, No, even though a 6th or higher level inquisitor with the Anger inquisition can 'rage like a barbarian' he or she does not have the Rage Class Feature, which is required for the feat (that said in my home game I would probably allow it assuming the inquisitor was high enough level to rage).
4. Yes, as far as I can tell.
5. Again, I do not understand the question.
Number 5 seems to ask what the default action is for spending ki or arcane pool points (write = written). The OP is wondering if the default action is swift.
5) It appears yes. The ability says "Each of these powers is activated as a swift action" when describing the default actions. I would assume that other cases that do not specify otherwise would be swift.