When you give a cat a crossword

Question

There I was, in the PSE lounge, polishing up my math-crossword. The only test-solver available was the Cryptic Cat. I think you see where this is going.

When I got my crossword back, the little bugger had thrown out all of my Across clues and replaced them with cryptics*. Worse, she** had gotten to my copy and destroyed my record of those clues. Then she had the audacity to say the crossword would be better this way. Well, that's for you puzzlers to decide.

Rules (given by the Cat)

• Across clues give words (or abbreviations) when solved. Each digit corresponds to a letter. Yes, you'll have to figure out the correspondence yourself.
  • There are 10 unique letters used, and they are consistent to their digit throughout
• Down clues give numbers when solved.
  • Clues that refer to Across answers should use the words' number values, once converted from letters
  • Most of the Down clues, if converted to letters, do not make words.
• No entries (Across or Down) start with 0. Yes, this means that no words start with the letter corresponding to 0.
• No Across words repeat, and no Down numbers repeat. In a few cases, however, an Across words' numbers (when converted from letters) are the same as a Down number.

Finally, you should anagram the letters used to answer the question: "What does the Cryptic Cat hate most of all?"

It is highly recommended to work with two grids, one where you enter letters and one where you enter numbers. Either grid (or both!) are acceptable answers.

Across
1. Note inside of clam (2)
3. Raise headless baby dog (2)
5. Helper, initially at back (2, abbr)
7. Expert security company’s outside episode (5)
9. Shake titanium ocean audibly (3)
10. A group of cops paged Borders (2, abbr)
11. Deal prince's lead deed (4)
13. Combines spots with one delta (4)
15. Computer is shot, heartlessly (2, abbr)
17. Applaud carbon loop (4)
19. "Pathetic loser" starts gaming company’s help request (4)
21. Special number detective (2)
22. Change current back (4)
24. Restaurant lied wildly (4)
26. Morning water drops off end state (2, abbr)
28. Left Hat: a powerful political coalition (3, abbr)
29. Overhear common way of getting four fancy skirts (5)
32. Rapper's calculator brand (2, abbr)
33. Whilst tailless viper (2)
34. Metal in epic ukulele (2)
Down
1. Product of 16 Down's digits
2. One digit is the sum of the others
3. 33 Across - 3 Across
4. Odd
5. 17 Across + 27 Down
6. Digits are odd
8. Digits average to 5
9. Prime
12. Divisible by 21 Across
14. 23 Down + 28 Across
16. 5 x 9 Down
17. Consecutive digits in some order
18. 2 Down x 75
20. Divisible by 28 Down
23. Divisible by 1 Down
25. 17 Down - 9 Down
27. Only one prime factor
28. Perfect square
30. 9 Across - 10 Across
31. Reverse of 30 Down

Text representation of grid. B indicates a black square, _ indicates an open one

1  2  B  3  4  B  B  5  6
7  _  8  _  _  B  9  _  _
B  10 _  B  11 12 _  _  B
B  B  13 14 _  _  B  15 16
17 18 _  _  B  19 20 _  _
21 _  B  22 23 _  _  B  B
B  24 25 _  _  B  26 27 B
28 _  _  B  29 30 _  _  31
32 _  B  B  33 _  B  34 _

_{* Thanks to @jafe for helping me polish these clues!
** We know that the Cat's a she because she recently had a litter. That's where all the recent cryptic clues have come from}

Answer 1

Clue explanations and solved grid:

The thing Cryptic Cat hates most of all is

DUPLICATES, as can be found by anagramming the 10 letters in the grid.

It's not actually necessary to associate letters to digits to get the final answer - it can still be done, though. Here's one of the many possible logical paths you can take:

16D and 18D must end in 5 or 0. The letters at the ends of those entries are A and I; since 0 cannot start an entry, and A starts 2D, we get A=5 and I=0.
Next, clue 18D can be examined in more detail. To have 5?? * ? = ?0??0, the second digit of the first number must be 4. So D=4.
From 4D, D is odd. From 9D, C is odd. From 8D, E is odd (because D is odd).
From 16D, 9D must be less than 20. So T=1. Since D and E must be two odd numbers on opposides sides of A, they are 3 and 7 in some order, and C=9.
From 17D, P=8.
Looking at 18D, 538*75 fits the pattern but 578*75 does not. So D=3, E=7, L=4.
From 5D, 27D must be 729. So U=2, S=6.