Is it normal for my eyes to get watery when I play a game?

Absolutely. Intense visual attention to books, computer screens etc. Will result in infrequent blinking so that tears are not spread across the eye to wet the developing dry spots from evaporation. Use artificial tears frequently to avoid this. Patients with parkinson's disease often have terrible blink rates and exhibit moderate to severe dryness of the corneas and conjunctiva.
Yes. Any activity that requires concentration can cause watery eyes. Staring for long periods decreases blinking. This in turn dries the eye's surface, irritating it and sending a signal to the tear gland to produce more tears. The tearing stops when the eye's surface becomes less irritated and you start blinking naturally.