I’m including this info because I believe something electrical is causing their occurrences.
Really?
1. Empty vase on counter slid about 12 inches. Checked the vase and counter and ensured that both were complete dry so condensation was not a cause.
2. A glass “flew” off the counter but didn’t break when it hit the opposite wall. When she picked up the glass and examined it she did not find any chips, breaks, fractures or scratches on it. She put the glass in the sink and walked away. When she came back to the sink and picked up the glass to examine it again, it broke in her hand but the breaks were in “U” shapes on opposite sides of the glass rim. The rest of the glass remained OK.
4. Her husband put a spoon down on the counter. While his back was turned he heard a scraping sound. When he turned back he saw that the spoon was pointing in the opposite direction.
How exactly could something electrical cause any of this? Out of six examples of "strange" activity you've prevented, at least half clearly have nothing to do with electricity, so I don't really see how your belief about it being power lines is any more reasonable that their belief that it's ghosts. Neither claim actually fits the observations.
It's also worth noting that a few hundred yards isn't actually very close to power lines at all. I live maybe 30 metres from some, and a lot of other people live much closer than that. If the stray field hundreds of metres from power lines was capable of destroying dishwashers and throwing glasses across rooms, you can be sure people would have noticed it before now. It also doesn't make sense that a few hundred metres away the field would be strong enough to do all this, but a few hundred metres plus the width of a room it isn't. So yeah, forget the power lines. Some of the things could be caused by electrical faults, but not by distant power lines.
As for more specific explanations:
1. Not enough information. Had the vase just been put down, or maybe just touched by something and we're supposed to be surprised it moved so far? Or did it just spontaneously move when no-one was near it and any movement at all is surprising? What kind of surface was it on? What's the base of the vase like?
2. "Flew"? As in it jumped into the air all by itself, or someone dropped/knocked it and was surprised how far it went? If the latter, I don't see anything surprising about this at all. Dropping isn't exactly a controlled action so things often end up where you're not expecting, and things don't always show obvious damage when dropped even if they are actually seriously weakened. I've had several instances of apparently undamaged glasses breaking, and I expect most other people have as well. Unless they're claiming it took off by itself, I just don't see anything interesting about this one. And if they are claiming that, the part where it later broke still isn't interesting.
3. I don't know how ice dispensers are supposed to work (why do Americans seem to be so obsessed with ice anyway?), but again there's more information needed. They shoot across the room in the sense that they're flying through the air, or in the sense that they come out slightly faster than expected and slide along the floor? As for why they come out at all, it could be a mechanical or an electrical fault. Has an engineer actually checked the fridge and tried to fix it, or at least explain what's wrong with it? If not, why not?
4. Again, needs more information. He put the spoon down and immediately turned his back and heard it spinning, or he turned his back and heard it a few minutes later? For the former, not really interesting at all. Sometimes things slide/spin/generally move around if you're not careful putting them down. For the latter, again not particularly interesting. I'm constantly thinking I placed things differently or even losing them entirely, even if I've only been holding them just a few minutes earlier. The exact placement of a spoon isn't the kind of thing people tend to may that much attention to, so the most likely explanation would be that he simply remembered it wrong.
5. Simple bad luck is an option, or there could be some electrical fault. If your description of the wiring in 6. is correct, the house electrics certainly sound rather dodgy. Lights are almost always on a separate circuit from appliances, so to have that not be the case in two different rooms suggests there could be some issues. Have they actually had anyone check it out? Did they even have anyone check out the second and third dishwashers?
6. Dogs being scared in a thunderstorm is not at all unusual. Lights going out in a thunderstorm isn't unusual. And the things you say are on the same circuit as the lights generally shouldn't be, so that's something to check otherwise the whole thing is completely unsurprising (also, it's recommended to turn off things like TVs and computers anyway during thunderstorms since their electronics can get damaged). As for the "dark shadow", the middle of a thunderstorm in a room with dodgy lights is going to be one of the best times for seeing funny shadows, since who knows what light sources there are going to be at any given time.
So yeah, obviously we can't give any solid explanations here, but some of these things just don't sound interesting at all, while the others really have far too little information available. And remember, it's no good asking for that information now, since they will be sure to remember things in a way that supports their beliefs. Not because they're dishonest or stupid, but simply because that's the way the human brain works - if you don't know all the details, it fills them in itself in a way that fits with what it already thinks it knows. The only way to get more information is to wait until things happen again and make sure to record them properly at the time. Preferably with video or pictures, but at the very least write everything down as soon as it happens. I'd definitely recommend doing that before getting people to wander around their house, especially since it doesn't seem to be particularly regular.