Letter Boxed validator
05AB1E, 37 35 33 32 31 29 28 bytes
εk3÷üÊ}DO2@¹ü«εüQO}²{¹˜êQ)˜P

-2 bytes by taking inspiration of the ê approach @Emigna used in his 05AB1E answer.
-3 bytes thanks to @Grimy.
Takes a list of list of characters for the words as first input, and the flattened list of twelve letters as second input.
Try it online or verify all test cases.
Explanation:

ε         # Map over the character-lists `y` of the (implicit) input-list of words:
 k        #  Get the index of each character in the (implicit) input-list of letters
  3÷      #  Integer-divide each index by 3
    üÊ    #  Check for each overlapping pair of integers that they are NOT equal
}D        # After the map: duplicate the resulting list
  O       #  Get the sum of each inner list of truthy/falsey values
   2@     #  And check that each is larger than 2 (so all words had at least 3 letters)
¹ü        # Get all overlapping pairs of character-lists from the input-list of words:
  «       #  And merge them together to a flattened list of characters
   ε   }  # Map over those merged character lists:
    üQ    #  Check for each overlapping pair of characters in the list that they are equal
      O   #  And take the sum of this (where we'd expect 1/truthy if the last character of
          #  the first word and the first character of the second word are equal)
          #  (NOTE: This could fail for inputs with identical adjacent characters,
          #   but the earlier check of `εk3÷üÊ}` already covers for this)
²{        # Push the input-list of letters, and sort them
  ¹˜      # Push the input-list of list of word-letters, flattened,
    ê     # and then uniquified and sorted as well
     Q    # And check if both lists of characters are the same
)˜        # Then wrap everything on the stack into a list, and deep flatten it
  P       # And check if everything is truthy by taking the product
          # (which is output implicitly as result)