Gash where valve meets tire tube; How could this happen, and can I fix it?
Most likely this is a result of riding the tire while the pressure is too low.  When you do that the tube tends to slide around in the tire and can get cut against the edge of the rim.

However, this could also be the result of using a hand pump, which, if you don't "buck" it against something or hold the head steady will move the stem back and forth rather violently.  A "floor pump" prevents this sort of damage (and also makes inflation much easier).

The tube is toast.  However, this is not a particularly expensive repair.  A new tube will run you $5-10, and replacing it is something you need to learn to do anyway.