You and your wife sound like the perfect candidates for a long-distance hike. I would preface with everything I say that planning and knowledge is very helpful. As your hike progresses, you will be surprised how well you adapt and your plans change. After several weeks on the trail, you will have figured out things 90 percent of what you need to know.
Given the parameters you specified, I think you will have little trouble adapting.
The NOBO permit into Canada needs to be submitted fairly early, if memory serves correct. But I would just submit a request with say, an Oct 1st timetable of getting across the border. It's a guesstimate at best anyhow. Nobody was standing there in 2009 when I crossed. Nobody ever asked for the papers, period, although technically you are supposed to turn them in.
The PCTA (www.pcta.org) issues permits for the PCT. They will issue a permit for any trip on the PCT of at least 500 miles, if you entertain a section hike. This includes the JMT if it is part of a trip at least 500 miles.
For the JMT, the earlier you can plan, the better. The trick is getting permits SOBO out of Yosemite. Northbound permits are easier to obtain, although I am not a fan of hiking up Whitney Portal. It's a slog to the top of Whitney and while pretty, it is a tough climb if you don't have any experience at altitude.
As far as planning and logistics here are my thoughts...
Starting May northbound....not an issue at all. It might be warmer, but a lot of people start a bit later. It can be hot some years, cooler others. I wouldn't worry about not having a lot of mountain experience, you will adapt. Just go slow the first couple of week and before you know it, you will be gaining miles. It's not a race, it's a journey, and the first 500 miles will feel like they take forever to complete. I guarantee when you get to the border to Washington state and see that it is about 500 miles to the end, you'll think, "No problem. A piece of cake."
As for food, you can largely buy along the route. However, you are NOT traveling through the part of California that believes in healthy food. The towns are generally small and have their share of places that deal in hearty (e.g. bacon-infused breakfasts), pizza joints and greasy spoons. If you are really picky, (or like me, a recent plant-based diet convert), it will be tougher. You can do it, but the further north you travel, the fewer opportunities you will have to visit big grocery stores near the trail. The question is how picky are you and will your tastes change? I will submit your tastes will change, and that's one reason why you may want to avoid mailing food instead of buying in town. You will likely need to mail some food to either post offices or stores/lodging that accepts packages in some instances (such as Kennedy Meadows in the Sierra).
On Gear? Well, shoes can be replaced pretty easily. Carry a smart phone and you can order online. Just be sure that you notify your financial institution/credit card company that they should expect charges along your route and that the shoe source / sporting goods place will ship to addresses that are not your home address. Some of these sites block this out of fear of credit card fraud. Generally speaking, the more established sites won't do this. Whatever you do, don't buy all of your shoes before your trip. Your feet may swell, this is common. I went from a 9.5 to a 11 during my trip.
Tips? Less is more. People generally pack far too much stuff; it's getting better but the amount of gear sent home after the first couple of days on the PCT (or the AT, for that matter) is staggering.
If it gets hot in SoCal and you are having problems adapting,get up early, hike until 11 a.m. or noon, find a place to hunker down near shade and water, take a break to 4 p.m. or so, and then hike the evening. We did this when temps routinely reached the high 90s / low 100s and am thankful I did. Evening hiking was much more pleasant. The mantra of 10 (miles) by 10 (a.m.) was common on such days.
I think the biggest issue people have is the feeling that they need to bust out miles right off the go. Just take it easy. You don't have to hike 20-25 miles right out the gate. If you can, great! If you want to go easier, not a big deal. Your body and feet will thank you for not stomping on the accelerator the minute your hike starts. The thing that will make you fall behind schedule are not these 15-mile days. Taking a lot of zero days (as in zero miles - rest days), while highly recommended, can and result in some consternation once the weather turns up north. You will have wished that you had not taken so many zeros and gotten north a bit faster, especially when greeted by cold rain or snow. (Honestly, September in Washington is generally fairly nice - but October can turn quickly). But zeros are fun, and necessary to recharge the batteries and give your feet a break. It is just easy to dawdle around and find that you need to make good time. (On this note, you will get faster as the trip goes on. MUCH faster. I got to the halfway point on Aug 1st. It took me more than three months to get there. It took only 2.5 months to finish the second half. And i was slow and dawdled.)
Most important advice - don't quit without giving it some thought. If you are having a bad day, or a couple of days, go into town and rest up. It's amazing how your view will change after a hot meal, a warm shower and clean sheets. Town at first can be great, but after a bit, I always found myself for pining for the trail again.
Have fun! My view of hiking long distances evolved a lot as I went up the trail. More than once I called home and told my wife, "IF I ever talk about doing this again, you can shoot me." But you know, now that I am done, all I want to do is another long distance hike! Hiking is a lot like life, you have your good days, you have your bad days, you have those epic days. You will forget the bad days and will generally carry sweet memories of the places you visited and the people you met along the way.