What if there was no minimum wage -- would someone pulling in $10 a month be above the poverty line in that case? The poverty line is the minimum income you need to get by, not the minimum you can pull in with a full-time job.

No - that's what is called a 'living wage' or sometimes 'family wage' to be clear that it should support a family and not just the worker. The minimum wage is in some places intended to meet that definition, but I don't know if the USA intended that when they created theirs, and because the federal minimum wage is only raised haphazardly instead of by indexing to living costs, it doesn't now meet it if it ever did (some states do index their minimum wage). I've heard people argue that minimum wage jobs are intended for teenagers who don't need to support anyone, for instance.

The poverty threshold, on the other hand, is based on actual costs - apparently it is "three times the cost of a minimum food diet in 1963, updated annually for inflation using the Consumer Price Index, and adjusted for family size, composition, and age of householder." [http://www.irp.wisc.edu/faqs/faq2.htm]

IIRC, the poverty line in the US is the cost of food, multiplied by three. Minimum wage can be below this, depending on region and number of dependents.

Poverty is defined by the means you can afford, and if that meets some rough threshold of basic necessities (food, shelter, possibly transportation, etc). In a perfect world where policy is defined justly and the economy can support it without repercussions, the minimum wage would be defined by the poverty line, no the other way around. In the real world, there's no fundamental reason why the minimum salary in an economy would pass any particular bare-minimum lifestyle threshold.

I believe the magic number three was derived from one study decades ago, that found that food accounted for one-third of poor people's spending on average. There is no reason to think it has remained constant, or that it is stable to begin with, but that became the formula used in the US.

No, because you can have different numbers of people supported by a single minimum wage job. A single mother might have to split her income among three people (herself and her kids), but an older woman who takes a job just to get out of the house is obviously not in the same category.