ACROSS
 1. <falsify> SPOOF = SPF outside _OO_
 4. NIGH TUB outside C_L = NIGHTCLUB <speakeasy?>
 9. TEE + TO TALLER = TEETOTALLER <abstainer>
10. <grass> RYE = R_Y _E
11. (.m)UNIS(t.) = UNIS <educational establishments>
12. S IDES HOW = SIDESHOW <not the main attraction>
15. <cardinal> ALEPH-ZERO = HZ inside PAROLEE*
16. <they produce French letters> DUREX = DU REX
18. MAN outside SO = MASON <bricklayer>
19. MA + KEW + AVES = MAKE WAVES <to rock the boat>
21. (GENUS + DOG)* = GUDGEONS <fish>
23. <fruit> PLUM = "PLUMB"
26. <throw> RUG = _R _U_G
27. (NICOTINE + TAR)* = INTERACTION <combined influence>
28. <eatery> BRASSERIE = ←EIRE← after BRASS
29. ←LUG← after MO = MOGUL <big shot>

DOWN
 1. <mandarin> SATSUMA = SATS U M_ A
 2. (INNER SOLE)* = ONE-LINERS <cracks>
 3. ←GOLF← = FLOG <strike>
 4. <skillful> NEAT = NET outside _A
 5. <house-breaker> GOLDILOCKS = OLD inside GI LOCKS
 6. TURN _E DD OWN = TURNED DOWN <rejected>
 7. <wood> LARCH = L ARCH
 8. B_E E_S WAX = BEESWAX <business?>
13. CHANCER IE S_ = CHANCERIES <diplomatic missions>
14. <it stops> TERMINATOR = (METRO TRAIN)*
17. ←G NIT + LOVER← = REVOLTING <disgusting>
18. MAG HR ←BE← = MAGHREB <Northwest Africa>
20. <influential> SEMINAL = ←(.l)L ANIME S(e.)←
22. ←GOD← MA_ = DOGMA <creed>
24. T_ RUE = TRUE <loyal>
25. <fleece> SCAM = ←MACS←