12-volt came along in 1956 or so. They still used the same coil as the 6-volt systems but they put a ballast on it of around 1.3 Ohms or so. The 6-volt cars didn't use a ballast with the late type can style coil. The can types came back in with the 8BA and had been improved over the types used in the model A era so they didn't need a ballast.
The old coil testers had a warm up switch position so that the coil could be tested under normal working conditions. I wouldn't say that they get hot. I'd just say that it is normal for them to be warm. To me, hot will burn your hand immediately. Warm, you can hang onto for a few seconds before it gets uncomfortable. A coil is like an electromagnet that gets turned on and off really fast. When you subject a coil core to electromagnetic forces, it warms right up. If you forget to turn your ignition switch off and the breaker points are closed, the coil can get so hot that it will be damaged over a short time plus the battery will go dead eventually. Under normal use, the coil just gets warm.