When I run the T in the dark, I see one of the coils throwing a spark from the top! What does this mean?

All the coils should have a little bit of spark at the points. This is normal. If the spark is big and blue it usually means the capacitor is failing.

Steven's answer is correct if the spark is at and across the points. If the spark is coming from anywhere else there is a fault. Damp could have entered and allowed a 'leak'. This can result in the wood becoming carbonised and the leak then continues even when dry. Or there is an open circuit somewhere in the secondary coil and the HT finds another route.