When 1 card gives a damage icon to another card is it a "normal" icon or an "edge-enabled" one? If it depends on the card, how do you tell the difference?

It depends on the card. If the icon is just an icon in the text of the card, that means a normal icon (example: Jedi Lightsaber). If the icon in the text is in brackets, that means edge-enabled (example: Yoda).

See Combat Icons sidebar on pg 21 for reference/example.