haaron,
modern turbo-prop engines are so tightly cowled that it is unlikely that any external sign of fire will be visible. As noted above, there is not likely to be any engine instrument indication of a fire.
Only a very brave person would decide not to take the fire warning as real and ignore it. As a generalisation, most turboprops will fly safely with an engine shutdown, provided the recommended techniques are used.
So, to answer your second last question, there will most likely be no other indications of an engine fire other than handle & bell.
To answer your last question. I would not hesitate to shut the engine down if the handle was illuminated and the fire bell sounding. To do otherwise is to me, allowing the tail to wag the dog, as it presumes that the warning is false and so I'll ignore it in case I shut a good engine down. When the thought process should be, I've got an engine fire, I'll shut the engine down, land and have the fault investigated.
The dilemma would be if one of, the fire handle illuminated, or fire bell sounded, but not both. In this instance one might consider the possibility of a false indication and depending on a whole lot of factors one migh not shut down the engine if over inhositable terrain. However if near a suitable airfield, one would possibly consider a landing and investigation.
With reference to KingAir false fire warnings, if memory serves me, the first action on receipt of a fire warning in a LingAir is to turn through 90° and see if the warning goes out. If it does and it was daylight when the warning sounded, it may be assumed that the warning is false and a result of sunlight triggering the sensors.