She received $10,000 last year from the bank that was executor of the estate, and another $10,000 when all was settled this year. Will she have to pay income tax on this money?
Most likely, no taxes are due(of course, you should always consult your tax professional). In 2011, there were no Estate Taxes due on any transferred assets under $5,000,000.
No, based on the facts you presented. Assuming that she inherited non-IRA accounts (which this sound like), then the answer is no.
FYI: If she had inherited an IRA, then she would pay taxes on any distributions she receives based on her age (under 59 1/2 would be considered an early distribution). If this were an inherited IRA, then she would need to receive a distribution based on a specific schedule tied to her age and actuarial life expectancy.
Call the bank and try to talk to the person familiar with the distribution, after reading whatever was sent your wife earlier.
May be. If it was qualified money yes, if it was not then a portion of what she received may be taxable.