Greetings. First, you can petition the court for an order compelling her to bring the children to the airport, but your best bet is to threaten that if she does not you will file a Motion to Show Cause, ask her to pay all your expenses and attorney fees, and be held in contempt if she does not.
Attorney fees are costly, but well worth it. I do not know of any low cost legal fees (you could try legal services, but they generally do not do child custody matters). Your court order should deal with the extra expenses, but she could always ask for more child support if what she is getting does not cover everything.
Finally, no you cannot claim any of your children as dependents unless you have them for the majority of overnights in the year OR your court order allows you to do so. Best of luck.
Janet L. Fritts
Attorney at Law
4101 Lake Boone Trail, Suite 200
Raleigh, North Carolina 27607
The response posted above is based upon the limited factual information made available and is not intended as a full and complete response to the question. The only reliable manner to obtain complete and adequate legal advice is to consult with an attorney, fully explain your situation, and allow the attorney sufficient opportunity to research the applicable law and facts required to render an accurate opinion. The basic information provided above is intended as a public service but a full discussion with an attorney should be undertaken before taking any action.