I know that if I'm going to continue to ride I will need a real bike, but will a better bike make a significant difference in my ability to ride hills?
A good road bike, drop bar or flat bar, will make a huge difference.

How much riding does it take to improve breathing? Am I just not doing enough or is it my emphysema?
Could be both. Don't push yourself too hard. Just enough to see improvement each week.

How hard is biking on the knees? As I've been riding more my one knee is starting to hurt a little.
Cycling is very easy on the knees. I had arthoscopic surgery on my right knee 3 years ago. The physical therapy my sports med doc prescribed was riding my bike on the trainer for 3 weeks. Then he let me out on the roads again. He told me I could ride till I was 100.
I'm 66 so I guess I'll be enjoying riding for the next 34 years.
If one knee is hurting it may be your riding position, saddle height or saddle position.
Make sure you get a good fitting at the bike shop when you get your new bike.