Hard 5x5 when there's so many unchecked letters
2d - I don't love exhibits as a container indicator
also not sure on the parses for 1d ddef? Turn up as in the phrase 'Turn up for the books'?
and 4a edit: Just got the reversal - nice :)
I think every-other-letter checked (including first and last) is pretty standard on this style of grid! But in any case don't worry, now that barred grids are possible I'll be using those for basically everything :)
"Exhibits" is right on the line for me - personally I think it's okay for a container, as is "displays", while "shows" (which some constructors use) is a bit too vague. But that's certainly a matter of taste.