What are the problems that can occur with a bad engine mount? How is the problem fixed, are there any other problems this can cause to other areas of the boat? engine itself, gear, shaft, etc?
Well I suppose the biggest problem could be that the engine falls to the bottom of the hull and tears a hole through it and you sink. Shafts can become misaligned and you can twist those off. The exhaust system could be torn loose. Excessive vibration of the vessel which could cause your favorite beverage to fall into your lap.
To fix a bad engine mount would be to replace it if it is broken or needs service badly. The most common issue that I've encountered with engine mounts breaking is that the engine was not aligned with the shaft and the mounts were installed with an impact wrench. Shims are our friends that do their duty silently between the engine mounts and the engine beds. I had a boat come in that had a bad wiggle to it (a shake is bad enough, but this thing was dancing to its own beat when it was in gear) and the owner had a couple of friends install his engine and they just put it down on the beds and hammered the nuts tight with their impact wrenches, snapped the front engine mount right off and that stress cracked the rear mount on that side. After replacing the mounts and aligning the engine the front mount was off the engine bed by a full quarter inch. With the proper alignment the boat was better than ever, you could stand a 1/4-20 machine screw on the heat exchanger while in gear and watch it stay in place underway.
Personally I am not in favor of soft engine mounts, I prefer to bolt everything solid to the vessel. This eliminates quite a few issues with misalignment and vibration. If you consider the fact that every time you adjust your throttle the engine readjusts itself on those little rubber mounts which automatically puts your drive shaft out of alignment, an easy example of this is sitting in your vehicle and revving the engine, the vehicle will roll some due to the force generated by the pistons whirling around inside the engine block. Cars have universal joints to compensate for this misalignment and while they do make them for boats, why add hundreds of pounds, decrease accessibility and add headaches further down the road? If your engine is moving around on soft mounts, you aren't getting all the power transmitted through your drive line to the propeller, and that's just bad math any way you add it up.
Just my two cents