Generating nicer diceware passwords without losing entropy
To get "nicer" passwords, whatever that means, the suggestions are typically: 
  
  …
  
  
  use a selection method but be aware you lose entropy (e.g. pick $1$ out of $16$ passwords at the cost of $4$ bits of entropy)
  


What you are doing is effectively ‘a selection method’ (more commonly known in the literature as rejection sampling) with a bound on the number of rejections.  For your method to work, you need (say) a word list 16 times the size of a word list for standard (no-rejection) diceware in order to attain the same entropy while giving the user 16 distinct choices for each word position.


  1 - Am I correct that generating a password in this fashion does not decrease entropy?


Provided each of your 16 word lists has the same number of distinct words, you are correct that this does not decrease entropy vs. doing standard (no-rejection) diceware on one word list.


  2 - What is the impact of overlaps between lists when $N &gt; 2$?


As far as the probabilities are concerned, nothing, provided (a) each word list has the same number of distinct words, and (b) you always accept exactly one of the 16 options presented.  As far as the user is concerned, they may be presented with the same objectionable word twice for one position.


  3 - Why is nobody doing this? This is easy enough to automate. Unless something above is wrong?


How do you decide how to partition the word lists?

If you have a 16x word list, you could just do rejection sampling on that word list and as long as you reject on average fewer than 16 words per position in the pass phrase, you get the same security with less user interface complexity.  For example, you could draw 16 words independently from your 16x word list, and ask the user to pick exactly one of them.

Exercise: Find the probability distribution on what you get by selecting 1 of 16 choices drawn without replacement for each word position.  (Independent samples as in the previous paragraph, in contrast, are ‘with replacement’.)  This avoids partitioning the word list.