1. Special state (7)
EXPRESS - double definition. The first as in express/special purpose.
5. Sculpture shattered (4)
BUST - double definition.
7. Leaders of think tank penning little message (5)
TWEET - (T)hink (T)ank penning little (WEE).
8. Marriage evidently over? Don't say another word! (4,3)
RING OFF - is this a double definition? I think the word play is cryptic - a wedding band no long required.
10. Mug in Christmas stocking (3)
ASS - in Christm(AS S)tocking.
11. Warming device, unit with cobra snakes (9)
INCUBATOR - anagram (snakes) of UNIT and COBRA.
13. One million and two hurt (6)
IMPAIR - ome (I), million (M), two (PAIR).
14. Accompanying sailor, attain goal (6)
TARGET - together with sailor (TAR) there's attain (GET).
17. Combine eg nitrate compounds (9)
INTEGRATE - anagram (compounds) of EG NITRATE.
19. Low figure in question, essentially (3)
ONE - in questi(ON E) ssentially.
20. Coming soon, formal wear for garment (7)
NIGHTIE - coming soon (NIGH), formal wear (TIE).
22. Runner beginning to run, one runs! (5)
RACER - (R)un, one (ACE), runs (R).
23. Dry river, save bottles (4)
BRUT - river (R) is 'bottled up inside' save (BUT - all save/but the last items).
24. More air circulating in wardrobe (7)
ARMOIRE - anagram (circulating) of MORE AIR.
1. Old custom in movement of suspect abroad (11)
EXTRADITION - old (EX), custom (TRADITION).
2. Exercise before second drink (5-2)
PRESS-UP - before (PRE), second (S), drink (SUP).
3. Calling for hospital department, sickly (9)
ENTAILING - hospital department (ENT), sickly (AILING).
4. Well-groomed tree (6)
SPRUCE - double definition.
5. Cake, central point rising (3)
BUN - central point - nub - rising upwards (BUN).
6. Brief attempt about right (5)
SHORT - attempt (SHOT) about right (R).
9. In America, serious achievement of brilliant student (5,6)
FIRST DEGREE - double definition. The first has to be the American 'serious' as in first degree crime as the English first-degree burn is the least serious/superficial.
12. Car part collected in Birmingham (5,4)
BRAKE DRUM - collected (RAKED) inside Birmingham (BRUM - informal name).
15. Rising in heat, chilli coolish initially on good Italian dish (7)
GNOCCHI - I had some trouble spelling this to be honest. Rising - so in reverse order - (I)n (H)eat (C)hilli (C)oolish on (ON) good (G).
16. Snapper arrived before tailless ray (6)
CAMERA - arrived (CAME) before tailless Ray (RA).
18. Line round head of ginger cat (5)
TIGER - line (TIER) around (G)inger.
21. Young child’s measure of spirit (3)
TOT - double definition.