DFT Frequency domain analysis and interpolation I have a 2 part question, one may be related to why I'm not understanding the other.

A while back, I remember some professor saying that for the $N$ point DFT frequency domain, the values for $k$ around $N/2$ (halfway point for the spectral domain - those around $\pi$) are considered high frequencies, and the end points $(0, N-1)$ (those around 0, $2\pi$) are considered low frequencies.  I am having trouble seeing this mathematically given the definition for computing the coefficients of the DFT.  

$$X[k]=\sum_{n=0}^{N-1}\left(x_ne^{\frac{-i2\pi{}kn}{N}}\right)$$

To me, it seems that the frequencies grow as $k$ increases right up until $k=N-1$, where the coefficients repeat due to periodicity.  Perhaps I am mis-remembering and this may be an incorrect assumption?

This leads on to my second question.  I have seen a few examples floating around online, where to interpolate in the time domain, one may pad in the frequency domain (running the IFFT after of course).  The examples I have seen seem to decide to pad the coefficients around the middle, i.e. for $N=4$ and a desired sampling of $2N$, [1 2 3 4] become [1 2 0 0 0 0 3 4].  Is there a reason why the middle is targeted and not the end like [1 2 3 4 0 0 0 0]?

If my first question is correct, then this simply pads the higher frequencies as 0.  I am aware that these zeroes add nothing to the frequency information as $X[k]=0$ cancels any effect during the inverse DFT.  But wouldn't moving any of the coefficients change some frequency information, i.e. for [1 2 3 4] becoming [1 2 0 0 0 0 3 4]

$N=4, k=2, X[k]=3$, becomes $2N=8, k=6, X[k]=3$

$$e^{\frac{+i2\pi{}2n}{N}}\neq{}e^{\frac{+i2\pi{}6n}{2N}}$$
A while back, I remember some professor saying that for the N point DFT Frequency Domain, the values for k around N/2 (halfway point for the spectral domain - those around π) are considered high frequencies, and the end points (0, N-1) (those around 0, 2π) are considered low frequencies. I am having trouble seeing this mathematically given the definition for computing the coefficients of the DFT. 


You can define the DFT however you like, as long as in effect the same frequencies are covered - remember, $f(x) = e^{jx}$ is $2\pi$-periodic, so that, for example, the sum over $k=0,\ldots, N-1$ is the same as the sum over $k=-\left\lfloor\frac N2\right\rfloor,\ldots,0,\ldots,+\left\lfloor\frac {N-1}2\right\rfloor$.


  The examples I have seen seem to decide to pad the coefficients around the middle, i.e. for N=4 and a desired sampling of 2N, [1 2 3 4] become [1 2 0 0 0 0 3 4]. Is there a reason why the middle is targeted and not the end like [1 2 3 4 0 0 0 0]?


Again, different definitions of the DFT lead to different forms for zero-padding here.