Across:
1 Floors (AMAZES) – a (A-) challenge for rats (-MAZE-) at top of stairs (top of ‘Stairs’ = -S) (AMAZES),
4 Shares (STOCK-) first (1st = -IST) (STOCKIST) place with plenty in reserve (STOCKIST),
9 South American abandons tourism (‘South American’ abandons ‘tourism’= haecan = AACHEN) in German city (AACHEN),
10 Hot (H-) wing (-ARM-) from takeaway (-LESS) (HARMLESS) is fairly mild (HARMLESS),
12 In the historical sense, old Greek (HERODOTUS) child-killer (Herod the Great ordered the killing of the innocents/children when Jesus was born = HEROD-) to (to) return (= reverse indicator) (to = -OT-) from America (-US) (HERODOTUS),
13 Magnificent in The Palace (REGAL) knocking back (= reverse indicator) beer (lager = REGAL),
14 Tenor (-T-) in Othello (Moor = MO-OR-) and brief (e) series (‘cycle’ without ‘e’ = -CYCL-) is leading (1st = -IST) (MOTORCYCLIST) Hell’s Angel (MOTOR CYCLIST),
18 Presumably, this is what Polyphemus will do (Polyphemus was a one-eyed giant = KEEP AN EYE OUT) to not lose sight of something (KEEP AN EYE OUT),
21 Hemp producer (ABACA) is a (A-) graduate (-BA-) from California (-CA) (ABACA),
22 Philosopher (NIETZSCHE) of note (N-) made-up (= anagram indicator) ethics (ethics = -IET-SCH-) covering Zoroastrianism primarily (‘Zoroastrianism’ primarily = -Z-) for the foundation (‘thE’ foundation = -E) (NIETZSCHE),
24 Worker (HAND-) joins women’s institute (Irish Countrywomen’s Association = -ICA-) for power (-P) (HANDICAP) and that’s a disadvantage (HANDICAP),
25 Daredevils dismiss deed (‘daredevils’ dismiss ‘deed’ = arvils = RIVALS) by competitive opponents (RIVALS),
26 Reading (R-) the last line (last ‘linE’ = -E-) by The Observer (the observer = -VIEWER) (REVIEWER) critic (REVIEWER),
27 Dope removed from speedboats (‘dope’ removed from ‘speedboats’ = sebats = BEASTS) produced by the wild ones (BEASTS).
Down:
1 American (A-) rewrites (= anagram indicator) anthem (anthem = -NATHEM-) – it’s a (-A) (ANATHEMA) thing of loathing (ANATHEMA),
2 It’s telling it like it is (ACCURATE) to a (A-) Cork (-C-) member of the clergy (-CURATE) (ACCURATE),
3 Some of the Orangemen deceived (some of the ‘orangEMEN Deceived’ = EMEND) by change (change = EMEND),
5 Change of organ (change of organ/transplant = TRANSPLANT-) for editor (-ED) (TRANSPLANTED) moved to pastures new (TRANSPLANTED),
6 One of those not in the picture is accused of this? (CAMERA-SHY) Not wanting to be shot! (CAMERA-SHY),
7 One (I-) finishing sentence (finishing ‘sentencE’ = -E-) in jail cell (cage = -C- AGE) (ICE AGE) in a major cold snap (ICE AGE),
8 Criminal (= anagram indicator) lets us (lets us = TUSSLE) exchange blows (TUSSLE),
11 Yes (-Y-), get ready to do business (OPEN) with Los Angeles (LA-) in California, for one, (California = state – STA- -TE) (STAY OPEN LATE) and trade well into the night (STAY OPEN LATE),
15 It’s about (RE-) vet (vet/inspect = -EXAMINE) (RE-EXAMINE) in study again (RE-EXAMINE),
16 Buries (CONCEALS) in shrouds (CONCEALS),
17 They don’t believe (ATHEISTS) there’s a car (model -T-) with bank job (heist = -HEIST-) in flipping (= reverse indicator) South Africa (SA = A-S) (ATHEISTS),
19 Pick up (GATHER) omen from hatemonger (‘omen’ from ‘hatemonger’ = hatger = GATHER),
20 Posed (SAT) for most of (y) those overseas (“navy” without “y” = NAV-) (SAT NAV) – it’s the talk of the town on TripAdvisor (SAT NAV),
23 All the letters (AZ) sent back (= reverse indicator) (AZ = ZA-) to this country (Ireland = -IRE) (ZAIRE) in Africa (ZAIRE).