There are plenty of "how-to" guides for troubleshooting tube radios on the 'net, and there isn't much point in reproducing them here for the Nth time. The process is pretty much cut-and-dried; if you get stuck on associating a specific symptom with a problem, this forum can help.
With the coil loose in the can, it's possible that one of the coil's solder lugs is shorting out to the metal can. If the plate side of the primary winding has shorted to ground, it has probably burned up the transformer. If one of the secondary sides has shorted to ground, it won't ruin the transformer but the set won't work.
The IF transformer primary winding is more likely to fail than the secondary since it is the one that carries a fair amount of current from B+ to tube plate. This winding can burn out and open up, or internally corrode (and open up). This usually can't be seen by looking at the coil, but it can be easily confirmed (or ruled out) by just ohming it. If the winding reads a few ohms, it is probably OK. If it reads zero ohms, it is shorted. If it reads infinite ohms, it is open.