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