"The northeast" is a region of the United States roughly corresponding to the circled area in this picture:
As such, you can say you lived in "the northeast," which would mean you lived somewhere in that circle, but nothing more specific than that.
"Northeast" can also be used as an adjective, but in that case you need to describe what it's northeast of, otherwise the description is meaningless.
So you could say you lived "northeast of West Virginia," which would mean basically anything in the top 90% of the circle in the picture above.
Or you could just say "I live northeast," which would simply mean you live somewhere northeast of where you are currently standing. Again, distance is not really implied, so depending on context you might mean that you live in the neighborhood a few blocks to the northeast or in a town 100 miles northeast of the town you're currently in.
"Northeastern" is always an adjective, so you cannot say you lived in "the northeastern." There has to be some noun for it to modify.
As far as the distinction between "northeast" and "northeastern" when both are serving as adjectives, that's harder to pin down. I would say half the time native speakers make no distinction.
Sometimes I think "northeastern" implies a bit more precision than merely "northeast." For example, in the right context, "the northeastern United States" might correspond to this circled area:
That might be particularly true if, for example, you were standing in Delaware (clearly part of "the northeast," the region) and wanted to describe that you live in upper Maine.
Confusingly, though, "northeastern" could also potentially imply a less precise description.
Maybe you live in Phoenix, Arizona, where it's 70 degrees F in the winter, and you're having a conversation about the colder temperatures in "the northeastern part of the country." In that case, "the northeastern United States" might refer to this circled area instead:
Here are some corrections for your sentences (except #6, which is fine as-is). Note that we need to use an article:
- I lived in the northeast US.
- I lived in the northeastern US.
- I lived in the northeast of the US.
- I lived in the northeastern area.
- I lived in the northeast area.
- I lived in the northeast part of the country.
- I lived in the northeast. or I lived northeast of [somewhere].
- I lived in the northeastern part of the country.
Note that there is a potential usage of #8 unmodified – i.e., "I lived in northeastern country" – that would be grammatically correct, but only if you use "country" to mean dictionary.com's fifth usage:
rural districts, including farmland, parkland, and other sparsely populated areas, as opposed to cities or towns
That kind of usage would be pretty folksy sounding to my ear, and probably wouldn't accurately describe living in the northeast region of the United States, not least of all because it is the most densely populated part of the US.