Your income is subject to tax by the state that you live in and the state that you work in. It doesn't matter where your employer is located. In your case, you live and work in Alabama, so your income is only subject to Alabama tax. You do not have to pay any Georgia tax unless you sometimes go to Georgia and work there.
1. Can I have them deduct Alabama taxes instead?
You should ask them to do that. They might not be willing or able to withhold Alabama tax, but they can certainly stop withholding Georgia tax. If they do not withhold Alabama tax, you will have to make quarterly estimated tax payments to Alabama. They should not be withholding Georgia tax, because you do not work in Georgia. However, if you sometimes work in Georgia, they should withhold Georgia tax, but only for the days that you work there.
2. Will I have to pay both Alabama and Georgia taxes?
No, as I said above, unless you sometimes work in Georgia.
3. If Alabama gives me a credit for taxes paid in Georgia, will that credit be equal to the taxes that I paid in Georgia?
Not necessarily. It depends on the relative tax rates of the two states. Alabama probably would not give you credit for more than the amount of Alabama tax on your Georgia income. So if the Georgia tax rate is higher you would end up paying the full amount of Georgia tax, but no Alabama tax, on your Georgia income. But if Alabama tax is higher, the credit would be equal to the Georgia tax that you pay, but you would still pay some Alabama tax in addition.
4. Can my wife and I still fill jointly even though we are earning income from two different states?
Yes, you can and should file jointly in any case. But you might not actually have income from two different states. If you don't work in Georgia at all, all of your income will be Alabama income.
