How can I get information about properties of a style?
You need
CurrentValue[{StyleDefinitions, stylename}]
Examples:
CurrentValue[{StyleDefinitions, "Section"}]
CurrentValue[{StyleDefinitions, "Section", "CellFrame"}]
(* {{0, 0}, {0, 1}} *)
CurrentValue[{StyleDefinitions, "Subsection", "FontFamily"}]
(* "Helvetica" *)
CurrentValue[{StyleDefinitions, "Graphics", "CapForm"}]
(* "Square" *)
etc.
This is just an addendum concerning screen environments. To obtain all the styles associated with an environment use a list as in:
CurrentValue[EvaluationNotebook[], {StyleDefinitions, {"Text", "Printout"}}]
{CellMargins -> {{49, Inherited}, {Inherited, Inherited}}, Hyphenation -> True, LineSpacing -> {1, 3}, TabSpacings -> {2.5}, CounterIncrements -> "Text", MenuSortingValue -> 1400, MenuCommandKey -> "7", FontFamily -> "Source Sans Pro", FontSize -> 15}
or to obtain a specific setting:
CurrentValue[EvaluationNotebook[], {StyleDefinitions, {"Section", "SlideShow"}, FontSize}]
40