This troper works for the same grocery company as Randy. The name tag he uses can easily be changed, as those badges are blank and filled-in with a label maker by managers rather than custom printed. I know the manager is a complete douche, but seriously, it's actually within the stores better interest to simply make a new sticker with his preferred name than keep it, due to: 1) customers frequently address employees by name and Randy's ego would cause him to constantly correct anyone who refers to him as "Robin", 2) suppose his direct supervisors/other employees know him as Randy; wouldn't they get confused if a customer asked/praised/complained about some guy named Robin? 3) are other people in the store named Robin? calling him by his preferred name (or even as Ram!) could prevent confusion over loudspeakers or incident reports.
Like you say, I think it's just emblematic of the store manager being a pedantic Control Freak tool.
The fact that Randy really doesn't like being recognized on the job might mean that he doesn't mind the name tag as much as it first seems, especially since fans might have no idea what his legal name is.
How the hell did Randy afford the rights to Sweet Child o' Mine?
Most (read: almost all) indy wrestling shows don't bother securing the rights to songs they use. Sometimes this does get them in trouble. For example, Metallica sued Paul Heyman for $5 million for all the times "Enter Sandman" was played for The Sandman's entrance.
The show might not have been televised. ROH did have a TV deal but it tapes a lot of its shows exclusively for DVD, as do a lot of other indie promotions. Technically indie promotions don't secure rights for the songs the wrestlers use as entrance music. Though I believe ROH's sister promotion SHIMMER edited out the entrances when it first started taping for DVD because they were sold in actual shops so they needed to edit out the entrance music.
Hell, Randy might be an in-story friend of Axel Rose just like the real-life Rourke is.