Cropping turns photo background black , transparency is lost.

You don't have to shut down the computer to remove layer. You will find a trash can at the bottom of the layers panel. Select the layer(s) then the trashcan or drag the layer(s) to the trashcan (drag and drop). All but the last layer can be deleted.

As for a single layer to have transparency.

It must not be a background layer(will say background and have a padlock). Just drag padlock to the trashcan to convert the background layer to a normal layer.

If the layer is a solid color (for example) You would have to select all (ctrl-a on windows or cmd-a on mac) or use one of the selection tools for just part of the layer.

At this point you can just hit the delete key to remove any color within the selected portion(s) to reveal the transparency checkerboard.

If you are using artboards - You can select the artboard above the layer(s) and then right click for a pop up menu and choose ungroup. (I think the normal group commands will work, but have not tried - ctrl-shift-g on windows or cmd-shift-g on mac to ungroup - again I did not test so it may not work)

You stated you imported a picure whad picture did you import and how which import feature?