That's quite a distance for the chain to move (from the outer to inner, then off the inner chainring). I've seen that happen in crashes, but obviously not your issue. For the chain to move during pedaling (meaning there is tension on the top of the chain between the chainring and cassette), it seems that either the front derailleur cage is moving inward, or the chainring is moving outward.
You should notice if the derailleur had shifted inward unintentionally because you'd have to push the shift lever to reposition it to the outside when you remount the chain, so I assume that's not your problem.
Have you tested the chainring (crank assembly) for side to slide play? That could occur if a crank arm has become loose and is working its way outward off the crank shaft. You should feel zero movement if you push the crank assembly from side to side.
Or, if the bearings in your bottom bracket are completely shot, the wiggling back and forth of the top of your chainring might be bad enough to cause the unintentional shift (when the top of the chainring tilts outward, the chain would bump against the stationary derailleur cage and be forced inward). That should also be easy to detect by trying to wiggle the crank assembly.
Regarding the noises from the rear when you are coasting, that could happen if your free hub is beginning to bind up (meaning it wants to continue rotating with the wheel instead of remaining stationary with the chain). Also, if your free hub continues to rotate when you stop pedaling, the chain becomes very loose on top and could bounce off the chainring (that does not seem to be your dropped chain cause, because your dropped chain happens when pedaling. On the other hand, if your chain got out of position when you coasted to a stop, then it might move even more when you start pedaling. Unless a free hub is worn beyond repair, that binding can be remedied with disassembly, cleaning and lubrication.
You might also check the rear derailleur arm that holds the two jockey pulleys to make sure it's moving freely. If the assembly is quite dirty, or if the spring itself is broken, the spring may not be able to properly tension the bottom of the chain and make it more likely for the chain to come off in certain conditions.
Without seeing it firsthand, it's tough to judge (which is why the response is so wordy). I do suggest cleaning the entire drive train, which will make it easier for either you or a mechanic to inspect and repair. Of course, I'll be interested to hear the cause when you've figured it out!