I have had this problem and thought it was the taste of tin-foil that I used to rolled up and put into the belly of the fish to stand it up in the oven. So, I was very interested to hear an interviewer on the radio 3RRR in melbourne, talking to a chef who mentioned fish, like Snapper, are susceptible to that metalic taste if they are caught and die too slowly. The slower they take to die, the more stress the fish goes through and the stress hormone accumulates in the flesh particularly around the belly. I think they said a pinker snapper is better than a pale one for taste.
Most likely, the taste you are referring to is oxalic acid, which has a bitter/astringent taste and is found in many green vegetables including spinach, Brussels sprouts, green beans, collard greens, etc.
Herbs such as basil don't contain a lot of oxalic acid, and I can't say I've ever noticed it there, but they do have some.
One important thing to note is, cooking green vegetables and herbs in water (i.e. boiling them) is a very effective way of removing the oxalic acid - but the most important part of that process is removing the vegetables afterward. If they're canned, or added raw to a sauce, then the oxalic acid is just going to stew.
Dried herbs likely contain lower oxalic acid content because it's been evaporated away along with the water.
If you're only finding this taste in leafy greens (and not green beans, green peppers, broccoli, etc.) then it's also possible that you're tasting iron. It tastes like... well, metal. Some of the foods highest in iron include shellfish and seeds (pumpkin, sesame, etc.). So if you're finding a similar taste in those, even just a hint of it, it may be the high iron that you find "nasty".
Best Answer
Kudos to your wife for giving it another shot!
Let me start with what I feel is the most important part of my answer: find a good fish monger and make friends. If the supermarket is the best you can find, so be it, but learn the name of the person behind the counter, ask about the fish, be interested. Ask what's just in, ask what's fresh. If the fish is frozen at sea, ask if you can get some from the freezer instead of the stock in the case that's been thawed and sitting all day.
Ask to touch and smell the fish. Look at the stock - do the whole fish have clear eyes? Does it smell like a dumpster? Getting the freshest, best fish will help you and your wife enjoy it more. Some good suggestions here on shopping for fish.
Now, on to your bullet points. Fish should never smell or taste "fishy". Fishy smelling fish is a sign of bad fish.
I find that a lot of my friends who say they don't like fish have had experiences with poorly handled or old fish. Fish must be fresh (or frozen at sea and handled properly afterward) to be good. I've stopped ordering fish in restaurants unless it specializes in fish (and I don't mean Red Lobster) or is well known for their freshness.
That said, some fish is certainly more strongly flavored than others. I found this nice chart here with a few varieties of mildly flavored fish. I do agree that flaky, white-fleshed fish is generally the mildest. Flatfish, like sole or flounder tend to be consistently mild. Tilapia has become an extremely popular fish lately.
However if you do find a good fish monger, talk to him about it and you should be able to find plenty of interesting venues: monkfish has been called the poor man's lobster, and I had some arctic char the other night that knocked my socks off.
As for why white fish tends to be less strongly flavored than meatier fishes, I can't really answer authoritatively. My guess would be that meatier fishes tend to be more oily and fatty, which equals more/stronger flavor.