1. Due to the souring relationship between the U.S. and the Phillipines lately, Is it "(unreasonable)" to deny travel with the child at this time, until relations between our countries get better?
This is going to have to be a judgment call. Since you did not define specific situations in the decree that would be unreasonable, what is reasonable to one person may be unreasonable to another. You will have to convince the judge that if the child goes to this place, it will be a risk to the child's safety, and if the judge is convinced of this, the judge will probably agree with you.
2. Her home is in the southern island of Mindinao, also the home of the Abu sayaf muslim extremist group, who kidnaps and executes foreigners. Is it also "(unreasonable)" to deny her international travel with the child to this region because of this terrorism threat/risk?
Same answer as to question 1.
3. Is it possible to insert a bond requirement of $20k before her traveling with the child, since she will be getting 60k out of our retirement account upon dissolution of our marriage? I do not think that is reasonable. Especially when it was not a provision in the agreement.
4.What is the legal interpretation of (unreasonable)? Usually someone is acting unreasonably if their actions are not based in sound judgment, not fair, not sensible. Different people can differ as to what is reasonable or what is unreasonable. Someone acting reasonably would not act in an arbitrary or capricious way.
5. If I refuse to allow her international travel after this agreement if finalized, what are her options?
If she believes you have acted unreasonably, she can file a motion for order to show cause to have you found in contempt of court, and have the court enforce the decree by requiring your consent. She could also file a motion to clarify the decree to have the judge better define what is meant by unreasonable, or have the judge tell her whether her request is unreasonable. If the judge agrees with her that you have acted unreasonable, that may be a substantial and material change in circumstances that could justify granting her legal custody or could warrant changing the requirement that she obtain your consent, but again, she'd have to convince the judge that you unreasonably withheld your consent, which I do not believe the judge would certainly do, if you showed the court that the child's safety would by put at risk by going to these volatile countries.
Does that help?