Lower bound for enumerating k closest pair of points
This is only a partial answer. 

I think it is not complex to show that if question 1 can be done in $o(n^2 \log n)$, then so can be question 2. So according to the comment by Willard Zhan in the OP's question, your first question should be open as well.

A reduction could be as follows: 

Assume your input to question 2 is $\{x_i\}_{1 \leq i \leq n}$. Then construct an input for question 1 as follows:

$\{T + x_i\}_{1 \leq i \leq n} \cup \{- x_i\}_{1 \leq i \leq n}$ for a sufficiently large constant T.

Then the answer to question 1 with these $2n$ points could be easily translated to an answer to question 2, as pairs of the form $(T + x_i, -x_j)$ in the answer to question 1 are easily differentiated from the rest, and correspond directly to the answer of question 2, taking into account that: 


$|(T + x_i) - (- x_j) | = |T + x_i + x_j | = T + x_i + x_j &gt; T / 2$
$|(T + x_i) - (T + x_j) | = |x_i - x_j | &lt; T / 2$
$|-x_i - (-x_j) | = |- x_i + x_j | &lt; T / 2$