Is it normal that when you get really tired your eyes start to tear up and run?

Yes. When your eyes tear up and you are tired (at the end of a long day for example) this is often an indication that your eyes have become dry throughout the day. Your eyes then over compensate by producing tears in excess which may cause your eyes to water.
Yes. Often, tearing is a sign of, paradoxically, dry eye. When your eyes are dry, it is either because your basilar tear secretion, from the accessory tear glands in the fornices of the conjunctiva, is either insufficient, or because the tears themselves are evaporating too quickly. When this happens, the dry eyes are irritated, sending a signal via the nerves in the cornea and conjunctiva to the brain.
Yes. Yes, it is normal. When your eyes become tired, your eyelids do, too. Therefore, blinking diminishes and your conjunctiva gets more irritated. That, in turn, triggers feedback from the lacrimal (tear) gland to produce more tear to lubricate the irritated surfaces.