Really impressive progress KTurner!
The two more common causes of knee pain for new cyclists are a too slow cadence ("grinding") and a saddle too low. So you have to be especially careful with both.
* Cadence too low:
For most riders, 80 rpm (on the flat) is a minimum to be efficient and friendly to your knees. You should aim for 90-100 and be a true "spinner". Don't worry about being more out of breath with a faster cadence, it's actually because you are going faster! Pedalling fast is a habit to train, at the beginning you will find it awkward and tend to bounce on the saddle, but after some time it becomes natural.
You can use a cadence monitor but it's not too hard to count (for example 10s and multiply by 6, or 15 and multiply by 4). Using your speed and a chart might seem a bit complicated, but it's also a very good alternative (find a long straight bit of road and ride at a constant speed, noting which gear you are on). You only need to do that a few times to get a feel of what the cadence is, after that you will "know" at what cadence you pedal.
You raised a very good point by saying your feet fly out of the pedals when pedalling at high cadence. I honestly don't think it is possible to pedal at 90/100rpm on flat pedals. As you have certainly spent some time on this forum, I'm sure you already know the answer: clipless pedals. I know it tends to scare a lot of people but they are very very safe! Avoid the old style toe clips, they are either useless (if loose) or dangerous (if tight), SPD pedals on the lightest tension setting are VERY easy to get out, much easier than toe clips!
Having only one good leg for stopping on is not really an issue really. All clipless users (that I know at least!) have a favourite leg they always use for stopping: I always use my left one, and my right shoe stay clipped in from the beginning to the end of the ride. I just instinctively lean the bike a bit to the left (like you do I'm sure) when stopping and unclip my left foot, I think I have NEVER stopped on my right leg at all.
In addition, as your ride is hilly, you will develop muscles that are currently unused by pulling on the pedals (in addition to pushing): better for your knee, easier, and smoother pedalling!
* Saddle too low:
This is very important as well for your knees. I don't really know what advice to give you regarding you finding difficult to get on the bike but you have to find a way to have your saddle at the right height to avoid hurting your knees. I'm sure someone here can recommend a method to determine the right height (knee bent at 20 degrees when at the bottom of the pedal stroke? is it right?)
Don't worry too much about using 'wrong' gear combinations, even if it's not ideal, modern transmissions cope with this without any problem. Just try to avoid the 'extreme' combination (smallest ring with smallest cog) and you'll be fine, apart from this one, feel free to use all your gears, they are there for that!!