Primary Test: It is extremely difficult for grass roots jewelers and gemologists to be able to identify hydrothermal amethyst or related quarts. The reason is that the labs have been turning our synthetic quartz for so long for electronics use that they have it down to a very fine science. True identification requires the use of an FTIR unit which is well beyond most gemologists budgets. Occasionally one can find a seed plate from the original growth structure which is indeed diagnostic. Or there may be “bread crumb” appearing inclusions or secondary inclusions. But in general the grass roots jeweler and gemologist will be able to separate natural amethyst and other quartz members from their hydrothermal counterparts.
Secondary Tests: None. If there are no seed plate planes, and no bread crumb type inclusions then no identification should be attempted without advanced testing. There are reports of crystal twinning indicators but these have not been sufficiently documented to be reliable for separation.
The one feature that you may see is the colors produced by hydrothermal synthesis can be totally unlike anything natural, as is the case of the blue/green below. Sometimes common sense and the knowledge of the quartz families can help in the identification. Not diagnostic, but a very good indicator.
Repair and Setting: Based on this being a true synthetic quartz no special care is required other than would be used with a natural quartz. Not recommended for heating, but cleaning in ultrasonic should not be a problem.