Although the cold water may help you reduce the dark circles of eyes because the cold water stimulates the eye blood circulation, it still has the disadvantage. As we know, there are more bacterium in the cold water than that of warm and hot water. Thus, it may increase the possibility for you to get infected in eyes. You'd better not use cold water to wash your eyes often.
It is true that washing eyes with cold water can reduce dark circle of eyes. Dark circle of eyes is caused by an impairment of blood circulation in your eyes. Cold water applied to the eyes is a good way for you to relieve fatigue and reduce dark circle of eyes. But if you really want to reduce dark circle of eyes and want to have a pair of attractive eyes, washing eyes with cold water is not enough. You should also do the following things: firstly, you should go to bed as early as you can. Secondly, it is better for you to cover your eyes with wet towel regularly, which will make your eyes feel better. Thirdly, do some massage for your eyes and use some brands of eye cream. If you keep doing these things, I believe you will own charming eyes without dark circle of eyes one day.
Yes, it's fine to do that. And you can use warm water as well. For most people, it's common for them to wash eyes and lids twice a day, there's nothing wrong with using cold water. It disrupts the natural tear film and makes your eyes dry. However, in general, using warm water is better because it has more energy and the molecules in hot water are moving faster than that in cold water.