In the attached image, the 14 gauge wire coming from the left and into the junction box, is from a gas stove the floor above. The 12 gauge wire coming from the right is connected to the panel and is no longer used. It would sure be convenient if I could just tie these two together instead of running wire all the way to the panel again. Is there anything wrong with connecting a 14 and 12 gauge together? The stove only requires 15amps so the 14 should be ok on that front:
You can do this, but you must make sure the breaker for the circuit is a 15A breaker, not a 20A breaker. It is also possible that someone in the future may look at the 12 gauge wire coming out of a 15A breaker and think "I can up that to 20A by just replacing the breaker", so labeling it properly on the panel is important.