Why double error correction and quadruple error detection cannot occur at the same time?
With a hamming distance of five, you can make one of two assumptions:

A. No word has more than four errors. Since you need five changes to get from one valid word to another valid word, and five errors don’t happen, every invalid word has 1 to 4 errors. 

B. No word has more than two errors. If you find an invalid word, then it has at most two changes from a valid word. To reach any different valid word would take at least 3 more changes, so there is only one valid word within two changes, so you can fix the errors. 

But you can’t assume both at the same time. If you find an invalid word, it could have two or four errors, and trying error correction would give you a totally wrong result. 

If you have a hamming-distance of 7, you can indeed detect up to four errors and fix up to two errors at the same time. A word with four or three errors cannot be made valid with 1 or 2 changes, so you can detect 3 or 4 errors without being able to fix them, and you can fix 1 or 2 errors.