In an ephemeris, is it the North Node that is shown so we automatically know the South Node is opposite in our chart?
South node is always opposite the north node, yes. Whichever house the north node is in, south node is in the opposite one (10th/4th, 8th/2nd, etc.), and whichever sign the north node is in, south node is in the opposite sign at the exact same degree. If the north node is at 3 degrees Leo, south node is at 3 degrees Aquarius; if north node is at 11 degrees Scorpio, south node is at 11 degrees Taurus, etc.
Since the natal chart is the reverse of earth directions, are houses 12 - 7 the southern part of the chart and houses 1 - 6 in the northern section?
Yes, but it's not south and north in quite the same sense as the four directions. The southern hemisphere of the chart represents the sky overhead--that is, what's above the horizon; in a birth chart, what was above the horizon at the moment of birth--and the northern hemisphere represents the sky below, that is, what was below the horizon at that moment.
I'm confused because some people claim to have NN in their 10th, for example, and someone else has their NN in their 3rd. Wouldn't one of them be wrong?
No. Some people claim to have their sun in the 10th, and others claim to have it in the 3rd... does that make one of them wrong?
Nothing is necessarily in the same house in any two charts. The nodes transit through the zodiac just like the planets do, only they're always retrograde, and they take about 18 1/2 years to make a complete cycle. And just like the sun, moon, and everything else, they can be in any house in the birth chart, even for people who have them in the same signs.