Am I able to put my hands in the water, when the heater is in, and turned on, or do I unplug it first?
I actually put my hands in while it's plugged in. I have touch it plenty of times too...but I'm sort of reckless. Really, the makers of the standard aquarium heaters must have considered that. If it burns you, it burns your fish, it burns the plants, etc... poorly though out. Give some credit, hahaha (just saying, no insult intended!)
Am I supposed to leave the thermostat knob IN the water, or OUT of it?
Can't answer - I don't use that type. But I'm 95% sure that should be in the instructions.
What if plants touch it? (gonna try to hide it with decor and such)
Nothing happens. It will not melt. Or anything.
What type of mistake would cause a fire?
Leaving it out while plugged in (ESPECIALLY on flammable surface, and possible on carpet [I've seen carpet burns]). If long enough a fire will ensue... although I cannot say how long is "how long." However, if you leave it out and it's plugged in, you WILL burn yourself because (people say:) it burns like bejeezus out of water.
Also, if you lower your water levels and the heater is still on (this especially goes for 100% submersible heaters) and the heater is partly out of water, problems may happen. It could affect the heater itself negatively (not function properly afterwards if done too many times). It SHOULD shut off once out of water, but I never take that for granted.