Not to put too fine a point on it, since overall your calculations seem fair and your conclusions reasonable: but at $18200 with $10300 in deductions and exemptions the person's AGI would be only $7900 and hence all of their taxable income falls into the 10% marginal tax bracket. Of course this neglects payroll taxes (you did as well) but also ignores the EITC so suppose these roughly cancel. This results in the tax burden being $790 rather than your calculated $2730. That $2000 is a huge deal to a minimum-wage earner. Your back of the envelope calculation in this specific area was off by 450%.
Yeah I figured I would be off by a fair bit off when it came to the income tax math. I'm not sure whether there would additionally be some tax burden due to NY state income tax and NY city income tax. You're right that $2000 is huge to a minimum-wage earner -- that's almost 11% more disposable income.
On the flip side, minimum-wage earners are more likely to be employed part-time instead of full-time, but only 10% and 2% respectively of those groups are paid minimum wage or below, so it's pretty tricky to come up with a good estimate here.[1]
My main goal was to illustrate that the poor spend disproportionately more of their income on basic needs compared to the middle class, and inspire some empathy when we discuss the economics of poverty.
