Can hours and hours of computer work give you dry eyes?

Yes. Take frequent breaks.
Blink Regularly. Intense concentration during near vision tasks is sometimes associated with less frequent blinking and resultant dessication or dry eye manifestations such as dry spots on front corneal surface of eyes resulting in eye pain/irritation, tearing in response to dry spots on corneal and/or blurred vision symptoms.
Yes. Yes. Dry eye is very common. I would try artificial tears frequently during the day, ointment at night, warm compresses 1-2 times a day, and omega-3 fatty acid supplements. If no improvement, see an eye doctor.