You need to report everything relate to money on your income taxes. This includes gains and losses. If you don't report this stuff you could get audited.
This depends on the situation. If you have lost your whole income tax check, that was uncashed, then yes I would definitely report it to the IRS and they should be able to issue you another one. If...
As a general rule it is a good idea to report all income tax gains or losses. Contact a accountant to find out more about what you need to do for your specific situation.
When you are dealing with gains and losses, there is always something that outweighs the other. Income gains are always better than losses, but losses can sometimes affect the total of the gross...
The refund check, as income - No - for federal (it was taxed when overpaid - tax being paid with already taxed money), but a State one, yes. It was deducted from federal income.