Your eye can water for two main reasons.
One is that it is dry. The nomal tear film may be decreased causing your eye to produce a watery tear that just runs down your cheeks.
The second is that the tear drainage ducts may be clogged. The ducts drain tears that you produce into the back of the throat. If the ducts are clogged, the tears have nowhere to drain but down your cheeks.
These two causes can easily be differentiated in the office. The treatment for each (artificial tears, punctal plugs, etc. for dry eye or probing open the duct for clogged tear drainage duct) depends on the diagnosis.
Eye twitching (called myokymia) can occur as a result of irritation (allergies, dry eyes, etc.). But the most common cause is stress. No harm is done and it usually resolved within a few months.
I hope this helps and thank you.