Question: What do the drummers do? It says they increase moral. What does moral equate to in attack/speed for affected soldiers? Seems like it makes them faster, but not 100% sure. What would be the best use for them?

Answer: 
Upgrading the drummer only increases the radius of his influence over soldiers. Without an upgrade he can affect all the soldiers that are physically touching him. First upgrade allows him to enhance the 4 soldiers in the 2nd row away from him in each direction (up down left right), but not the soldiers in the corners of the 2nd row. The 2nd upgrade allows him to enhance ALL soldiers within 2 rows of the drummer in every direction. 

I still don't know whether or not it's effective from a damage standpoint to use that space on a drummer rather than an upgraded soldier, but it definitely depends on how many soldiers your drummer is touching and enhancing. If he's only touching 2 or 3 soldiers I doubt that it's worthwhile to waste that space. You definitely want to make sure your drummer is enhancing as many soldiers as possible to maximize the use of that square. I rarely upgrade the drummer because it's very seldom that there's enough contiguous open ground that you can completely surround the drummer in every direction for 2 full rows, or even 2 rows just in 4 directions. I usually arrange my drummer in the middle of 5 or more soldiers, sometimes as many as 8 if things work out that way.
The Drummer can increase the morale of soldiers with 30 or 40 percent. And when it upgrade to the third stage, it can covers more soliders.