What you are looking for will require a fair bit of compromise if you only want just one boat. Tandem canoes that can easily be paddled solo tend to be not that good for either solo or tandem paddling, thought that’s not true for every type of solo or tandem use (and “type of use” includes the size and nature of the paddler/paddlers himself/themselves, and smaller, lightweight people have a big advantage when it comes to “pocket tandem” canoes). It’s true that there was a time when lots of people solo-paddled tandem canoes, but that was before dedicated solo canoes became so widely available, and even then, solo paddling of tandem canoes was never done well by anyone other than experts, so for most of us, solo paddling in canoes is best done in boats made for that. To do all that you wish, ideally you would have two boats, and perhaps one of the two boats could still be a kayak.
The question of double-blade paddle versus single depends mostly on how much effort you wish to put into learning good technique. Single-blade paddling isn’t something you just pick up naturally in a short time. I made up my mind from the start that the single-blade was what I wanted to use, but it was very frustrating at first and it took me a whole year of frequent, dedicated practice before my technique was even remotely efficient, and another year to get reasonably good at some of the more nuanced stuff. Years later, I’m no expert, but to most people at least, it tends to look like I know what I’m doing Of course, it takes very little time at all to learn crappy single-blade technique that gets you where you are going in simple conditions, and for a lot of people that’s enough. For even more, the double-blade is the only method they ever consider for solo paddling. It’s all up to you.
There are a whole host of solo canoes that would fit the bill for solo paddling (though I’m leaving out consideration of the dog since there’s no indication yet regarding its size and temperament), and a few small tandems which may or may not be pretty good, depending on the type of use and paddler size, such as the Wenonah Solo Plus or the old Bell Morningstar. Sometimes the best thing is to decide which is your primary purpose and get some kind of boat that fits the bill for that, figuring that you can switch boats or add boats later on. That’s a case where you might see something for sale, and come back here with a question about using that boat for a particular purpose. Otherwise, the list of recommendations could be nearly endless.
I agree with Bob about strainers and boat control, and would add that learning to understand current falls under “boat control” too. For most situations where someone ends up in a strainer, the problem should be recognizable well in advance of when it actually happens, and if your wife thought of it in that way instead of thinking of it as an awful thing that happened, it probably wouldn’t be something that keeps her off the water now (though that gets into stuff that’s well outside of the obvious topic).