Wow, that's a lot of questions, all good ones.
I believe very strongly that it is beneficial to pedal with more leg muscles. The more one spreads out the work, the longer the muscles last. If one's bike fit is good, it will allow most any riding modality. Bike fit is a compromise. As your fitter pointed out, the trick is to have the compromise tilted toward the riding which you will do on that bike. What you are talking about is learning to "pedal circles." It's actually easier to pedal circles in a more upright position than on the drops, for instance. I think your fitter has it backwards about that. The main reason given for the common more upright touring fit is that one doesn't need to be as aero, as speeds are lower, so why not sit in a position that allows better breathing and leg mechanics? Yeah, tell that to the tourer who's battled headwinds for a week. Personally, I like the same position for everything except TT.

Most people find a lower Q factor (feet closer together) to be easier on the legs. Depending on how one's feet sit on the pedals, and one's bike geometry, one may have to watch for heel strike on the chainstays or crank. Personally, I pronate slightly when I walk, but have trained myself to bike with feet straight fore-and-aft. I find that kinder to my knees.

It's faster to pedal knees in. Aero testing tells us that pedaling knees in gives the same drag reduction as a set of high zoot racing wheels. However, not everyone's knees take kindly to that. It's generally recommended to keep the knee centered over the foot. That said, I know a number of long distance cyclists who pedal knees in all the time, even climbing. I also know some who pedal with knees way out. So it's all in what you got in the way of knees. One thing to watch is to try not to have side-to-side movement of the knees during the pedal stroke. That seems to be associated with some overuse syndromes.

In thinking about a good pedal stroke, think about applying a constant torque to the bottom bracket. This means always having power coming from one foot or the other, or both at a reduced force, all the way around the stroke. I find it best to concentrate on pulling the foot back at the bottom and then back and up for maybe another 30°, until the ham becomes ineffective. Your other foot will carry the load while that one comes over the top, so just take the weight out of the pedal, maybe push forward a little as it comes over the top. One doesn't need to concentrate on pushing down, as that seems to happen by itself.

There are four drills that make all the difference in climbing for me:
1) FastPedal. One warms up for 15' in zone 1, then pedals at a very high cadence, 115-120 if possible, however at a heart rate no higher than the top of zone 2, meaning that one can still speak in complete sentences but will breathe deeply and sweat. One does this continuously for 15-45 minutes, no break. One uses very low gears for this, setting the gearing with high cadence as first priority. If one can attain the cadence but effort is too low, one can increase the gearing. If one is out of shape, this may limit one to a max cadence of 100. Just keep at it, once/week. This almost has to be done on the trainer or rollers as it's hard to find the right road for it.
2) One-legged pedaling. Again on trainer or rollers. Pedal for 2 minutes with one foot, then the other, then legs together. Repeat for 15 to 45 minutes, continuously. Same warmup as for FastPedal. I like to pedal 50-55 and 80-85 cadence in alternate sets. The chain must remain taut the whole pedal stroke. If it goes slack on the backstroke, stop with that leg and switch to the other.
3) Low cadence climbing. Find a 10 minute hill and climb it at 50-55 cadence in the tempo zone (zone 3). Effort hard, but not extreme. No upper body movement at all, no prying on the bars. 3-4 repeats.
4) High cadence riding in easy rollers. Find a section of slightly rolling road about 30 minutes long. Ride it in the tempo zone at least at 100 cadence. Hold the cadence and effort as even as you can, uphill and down, the whole way. You'll have to shift a lot. Take 5 minutes of easy pedaling and head back, same thing again.

Notice that all of these drills are about becoming more efficient. I find that most people can get plenty of hard hills in their normal riding, so focusing on drills like these listed is a huge help. In my early season training, I do a few weeks of each of these drills, one once a week for maybe a month, in the order listed. Actually, since I start my training year in November, I get more than a month of the first two. Other than these drills, do normal, moderate rides, say 20 miles each, during the week as your recovery allows.

If you want to train for a distance ride, now is well into the time to start riding distance, once a week. 10% is plenty and compounds faster than you'd like it to. Just ride it moderate, zone 2, maybe a little zone 3 on the hills, but hold it down. That doesn't mean ride easy. Zone 2 is not easy, just well below the effort that produces leg pain. By now, you should be riding 60 miles or so on your long ride. Very gradually, increase your pace on the hills. About the last 6 weeks before your event, start doing the harder intervals or pass repeats, depending on the nature of your target ride. Depending on the length of your target ride, you should eventually be riding at least that ride length in total mileage per week, with a maximum necessary weekly mileage of 180-240, depending on your recovery ability.