I think these are correct but I need some help with the parsing......
1a. Island a century or so ago banning travel (7) - CORSICA (Island) Why 'banning travel'?
13a. Doesn't really appreciate facility for producing salary info closed ((4,3,7,2) - PAYS LIP SERVICE TO (Doesn't really appreciate) I can see 'Pay Slips' but not much else..
23a Watch casing light, mostly, and plain (5) - LLANO (Plain) I think it's 'Watch' = LO around (casing) most of a word meaning 'Light' = LAN but I can't think what word it is.
29a Record court business endlessly in it for long periods (9) - SEPTENNIA (Long periods)50a Quickly clear couch (7) - EXPRESS (Quickly)
56a. In spite of everything quits the case (4,2) - EVEN SO (In spite of.....) I think it's Quits = EVENS but that leaves 'O' for 'Case'?
4d. Dismay when apprehending inhuman action - unusual American feature (11,9) - APPALAICHIAN MOUNTAINS (American feature) I thought it was 'Dismay' = APPAL + an anagram of INHUMAN ACTION but I'm left with 'A' I can't account for.
28d. Delivered notes - supplied prompt (7) RESCUED
(Delivered) if 'Prompt' = Cue or Cued then why does 'Res' = 'Notes'?
30d. Where soldiers aren't pushed to dispose of game adversaries for the record (5) - ALBUM (Record) I can't parse any of it. (PS. 'Aren't is in italics).
38d. Involved in Times publication popularly - a series of books (9) - BYZANTINE
46d Entertainment featuring dancers with tops off by the way (3,4) - ICE SHOW (Entertainment)