The completed crossword looks like this:

Wait, what's going on here? Well, it turns out that each number in the grid does not represent the starting square for each clue corresponding to that number - instead:
...it represents a square in the grid where the equivalent Roman numeral must be positioned to help complete the word with the same clue number.
Note for starters that each numbered clue indicates a word containing as a substring its own clue number as a Roman numeral:
1. Scratched (6) = NICKED
2. Snow sport (6) = SKIING
3. Maui and Oahu? (6 7) = HAWAII ISLANDS
4. Pimento container (5) = OLIVE
5. Celebrated singer (4) = DIVA
6. Guest (7) = VISITOR
7. Old world warblers (9) = SYLVIIDAE
8. English king, briefly (6 4) = EDWARD VIII
9. Stick (5) = AFFIX
10. Cutting tools (4) = AXES
11. Left (6) = EXITED
12. Moving slowly (7) = TAXIING
13. Castle-dwelling neighborhood puppet (4 6 4) = KING FRIDAY XIII
14. Self-referential (9) = REFLEXIVE
15. Document for the IRS, perhaps? (3 7) = TAX VOUCHER
16. Tennessee city (9) = KNOXVILLE
(NB These words can also include other Roman numerals - e.g. DIVA contains 'DIV' (504) as well as 'V' - but only those which correspond to the clue number are considered for the placement of the words in the puzzle...)
Once these are placed in the grid as in the image above, the yellow squares can be rearranged to spell the word:
SIXTEEN, where the '10' space provides the Roman numeral 'X'. Since this is the total number of clues in the grid, and each clue refers to its own number, this is most definitely a self-referential crossword!