Q:

Five Letter Cryptex Code

Today I gave my three-year-old brother the five letter password to my cryptex and let him play around with it while I was in the other room. By the time I returned, he had managed to take apart the cryptex, change the password, and scramble it again. He couldn't remember the password, but luckily he had written down a bunch of things about the password before he scrambled it.

The key's fifth letter is alphabetically four letters away from the key's third letter.
The key contains only two adjacent consonants and no adjacent vowels.
Both the second and third letters of the key are adjacent to a consonant.
The third and fourth letters of the key are opposites (vowel/consonant).
The key’s fifth letter is “I”.
The key’s second letter is alphabetically seven away from the key’s fourth letter.
The key’s second letter is alphabetically eleven away from the key’s third letter.
The key’s first letter comes alphabetically five before the key’s second letter.

The password is five letters long, contains only letters, and may not even be a word.
Can you help me regain access to my cryptex?

A:

We're given this:

The key's fifth letter is alphabetically four letters away from the key's third letter.
The key contains only two adjacent consonants and no adjacent vowels.
Both the second and third letters of the key are adjacent to a consonant.
The third and fourth letters of the key are opposites (vowel/consonant).
The key’s fifth letter is “I”.
The key’s second letter is alphabetically seven away from the key’s fourth letter.
The key’s second letter is alphabetically eleven away from the key’s third letter.
The key’s first letter comes alphabetically five before the key’s second letter.

We find:

 _ _ _ _ I — we're given this from #5.
_ _ _cI — 4th letter cannot be a vowel, from #2.
_ _v cI — 3rd letter must be a vowel, from #4.
_ _ EcI — from #1 (must be -4, as +4 is "M" and is not a vowel)
_ P EcI — from #7 (unless "alphabetic distance" is mod26?)
K P EcI — from #8
K P E W I — from #6  (must be +7, as -7 is "I" and is not a consonant)

This appears to meet the criteria given.
We can confirm the conjecture about alphabetic distance not wrapping around must be true, as otherwise ...

 #7 would give T for second letter, giving O for the first letter, and O T E W I then does not fulfill #2.

