Suppose my salary for the year is $1000. My tax bracket is 10%, thus my employer has withheld $100 and I only ever get to see $900.
I'm going to just pretend these assumptions are compatible and answer theoretically, glossing over the details of what would actually happen if you actually earned only $1000 in a year. In reality, if you made only $1000 per year you would not owe any tax.
Question #1: I would like to put everything I earned ($1000) into a traditional IRA. Since you put "pre-tax" dollars into a traditional IRA, I should be able to contribute all $1000 to my IRA. But how can I if I only ever get to see $900?
You could file a W-4 with your employer, claiming extra exemptions to reduce your withholding. If you plan on claiming adjustments to income (such as IRA contributions) that would substantially reduce your income, you could in theory reduce your tax withholding to zero.
You can also do as you asked about in your Question #2: contribute your $900. Your tax liability will and you'll get a refund of the tax that was withheld (or part of it, depending on how much income you still had after your IRA contribution).
Question #2: Suppose I only contribute $900 to my traditional IRA for the year. Is my Adjusted Gross Income now $100 (i.e. $1000 - $900)? Does this mean my income tax is $10 (10% bracket) and I should get a $90 tax refund? Yet, this seems unfair that I owe any tax at all, when I didn't see any real income at all because the $100 was withheld by my employer all year. Is this really how it works?
Imagine the money had not been withheld, and you had received the $100 in cash. You would then owe $10 in taxes, leaving you with $90, which is the same amount you got from your refund. Either way you wind up with $90.
It is true that you cannot contribute the whole $1000 because some is withheld. You can get around that by increasing your allowances with a W-4 as I explained above.
Question #3: Suppose I get a $90 tax refund. This refund is "after tax" right? That is, taxes have already been paid on that income (the $10), and $90 is what I got to keep after taxes albeit in the form of a refund. If I now go put this $90 into a traditional IRA, then when I withdraw from my IRA later on, will it get taxed again?
Withdrawals from a traditional IRA are taxed as ordinary income, so yes, you will pay taxes on it when you take it out. However, in the year that you put it into your IRA, you will lower your taxable income by $90, reducing your tax in that year.
For instance, suppose that in the year 2016 you earn $1000, contribute $900 to your IRA, and get a $90 refund, as you described above. Then suppose that in 2017 you again earn $1000, but now you contribute the $90 from last year's refund to your IRA. Since the refund doesn't count as taxable income, you taxable income in 2017 is only 1000-90=$910, although you actually earned $1000. This means you'll only owe $91 in taxes, not $100. So, although you will pay taxes on the $90 when you take it out, it saved you money on your taxes in 2017, when you put it into the IRA.
In practice, you will never be able to get things to work out so that you end every year with the exactly correct amount of tax having been withheld. You will always either get a small refund or owe a small amount of extra tax (or, even worse, get a large refund, or even worse, owe a large amount of extra tax). You can't expect your tax payments and refunds to completely balance out over any fixed number of years. Inevitably, there will be a slight mismatch which will carry into future years. In theory, you won't lose any money due to this sort of juggling. (In practice, you may lose money, but it won't be due to the juggling per se; rather, you can lose money due to being in different tax brackets at unfavorable times, or due to changes in tax law that come at unfavorable times.)