I think I agree with you. I've always just thought of infinity as having no end - and that's helped me understand all these paradoxes. For any N, 2N is certainly bigger, even if they're both so big I could never measure either. In my mind, infinity follows all the rules of mathematics that other numbers do (i.e. multiplying inifinity by two ensures that it's even, IMO), it's just that it has no end. Parallel lines don't meet at infinity - they have no meeting point.
When you write 2N do you mean "2 times N"? If you do, then your intuition is at odds with mathematics as generally practised, because 2*aleph_0 is aleph_0. Further, two times infinity is the same as three times infinity, and there is no concept of infinity being even or odd. Claiming that 2 times infinity is even can lead to contradictions and inconsistencies.
From that point of view, it would appear that your intuitions are not helping you. More, under some models of geometry it makes a lot of sense to say that parallel lines really do meet at infinity. It makes some theorems a lot easier to state, and easier to prove.
So you may actually agree with a lot of what I'm saying, but your arguments to support that claim appear, at least on the surface, to be wrong. It may be that you have some understanding of these "paradoxes," but your other statements suggest that your inderstanding is not that of current or classical mathematics.
