1) The first question is, if I receive this refund to my US bank account, do I need to pay taxes on this "income"? If your tax liability was deferred at the time you made contribution into your IRA account - it is your taxable income at the time of distribution regardless how distributed funds are used. So - for income tax purposes the amount of distribution is recognized as your taxable income and you should receive the reporting form 1099-R from the payer.
2) if I receive this refund to my Israeli bank account, do I pay tax, and where? If funds are transferred into US bank account or abroad has no effect on income tax obligations.
3) Can I report the part of the donation that isn't being refunded as a tax deduction since my only proof of the donation is a wire transfer to the bank of the organization in Israel? You may deduct the amount you gifted to qualified charitable organization. o become a qualified organization, most organizations other than churches and governments must apply to the IRS and receive a determination letter.
4) Because the branch is itself in Israel, can I still claim this donation since technically the donation was made to an "International" company?
You can ask any organization whether it is a qualified organization, and most will be able to tell you. Or you can find it on the Internet at www.irs.gov/app/pub-78. You can also call the IRS to find out if an organization is qualified. Call 1-877-829-5500. (For TTY/TDD help, call 1-800-829-4059.)
As long as that is a qualified organization - you may claim a deduction.
You may be able to deduct contributions to certain Israeli charitable organizations under an income tax treaty with Israel. To qualify for the deduction, your contribution must be made to an organization created and recognized as a charitable organization under the laws of Israel. The deduction will be allowed in the amount that would be allowed if the organization was created under the laws of the United States, but is limited to 25% of your adjusted gross income from Israeli sources.
If the refund or other recovery and the expense occur in the same year, the recovery reduces the deduction or credit and is not reported as income.
If you receive recoveries that were deducted in previous years - there are relatively complex rules how to determine taxable part - see for details IRS publication 525 page 22 - http://www.irs.gov/pub/irs-pdf/p525.pdf