Frequently washing the inside of the eye by splashing water in it is not a good idea. Tap water may be unclean. Even RO water (purified) is not recommended as it does not have the same composition physiologically as tears, nor the same PH. Tears are made just for the purpose of keeping the eyes moist and do their job well. Only if you have dry eyes (your doctor will diagnose this for you), you may need supplementation of your natural tears. For this there are quality controlled pharmacological lubricant eye drops.
Rubbing your eyes for any reason (itching, discomfort, in order to see better) is not recommended at all. You may feel a sense of satisfaction, but you are actually harming your eye. By rubbing your eye, you can accidentally graze the surface of your cornea, specially if there is a small foreign body in your eye. Rubbing the eyes can induce inflammation and congestion. Frequent rubbing is also a strong contributor to a disease called Keratoconus where the cornea becomes thin and bulges. What you can do instead when you feel like rubbing your eyes, is to put preservative free lubricating drops (available over the counter) for soothing the eye till you get further treatment from a doctor. You can also hold ice in a cloth for a few seconds over your closed lids for relief from itching (holding it for too long can cause pain).