I, too, have used headphone jacks with devices with success. But I'd generally advice against it. In my case I suspect, as many effect devices expect a high impedance input, and headphones out are low impedance, that it had to work too hard, got overloaded and hot, until it just died. It "worked" for a couple of hours, but with strange behaviour, hangups etc.
Never heard that one should leave a synthesizer on for such a reason, sounds really strange. Especially if it's not behaving correctly, take the analogy of the car, I think it is applicable to electronic devices as well. If a component is broken, it would affect the circuit, perhaps loading other components in ways it wasn't intended to. If my stuff doesn't work properly, I turn it off immediately and have it fixed ASAP, or unplug everything and troubleshoot step by step.
I don't know what's happening, that's why I encourage thorough troubleshooting, unplugging everything and then plugging in bit by bit, to isolate any eventual causes. You mentioned it worked like it should before, when nothing was connected. If it, then, doesn't work properly when connected to other stuff, I'd suspect it's not the device itself but some connections that may cause this.
I've tried some complex setups myself, but almost always ended up with ground loops. Now I try to use as few devices as possible, connected to the same outlet and audio interface, and a dedicated midi interface. Connecting to different outlets also invites ground loops, like you mentioned using an extension cord from one room for some device. I'd try to use the same outlet for everything, and as few branch jackboxes (I still don't know the english word) as possible. Avoid daisy chaining. Keep it as simple as possible. DI boxes can also be helpful with ground loops. USB invites ground loops way more than MIDI, so if necessary I'd use a high quality powered USB-hub, cheaper ones are most often a source for ground loops.