Let's say you make $10,500.00 a year in a made up country. Anything over $10k is taxed at 20%, so your net income is $8,100.00

If you make $10k or under, you are taxed at 10%.

So really she'd get $9,400 since taxes only apply to that over the bracket. $1,300 more than you claim.

If you donate over $500 to charity, you receive an additional 2.5% tax break.

Which is ridiculous and no country does this.

$8,100.00 vs $9,260.00

Neither figure here is right. At the 10% < $10k, 20% > $10k she'd earn $9,400 no $8,100 and $10,000 - 7.5% is $9,250 not $9,260.

As great as charity donations sound, they are not always in good faith.

True, but not for any of the bad maths you've demonstrated. The major problems with donations is when someone donates something valued more than it's true value in order to offload something for a larger tax writeoff than it's true value(normally done with art: buy a piece of artwork for $5,000 get it valued at $50,000 and donate it for a $10,000 tax writeoff netting you $5,000 in profit; or items that don't hold their value, buying something now, enjoying it for 10 years, then donating it claiming it's original purchase value as it's value but really if you sold it you'd only get a fraction of that, the tax writeoff is then worth more than you'd get selling it).