That's pretty simple. You get faster by riding faster and stressing the systems which are slowing you down. 2x10-20 minute intervals at a threshold pace once then twice a week will do wonders with the other side of the equation being rest days (at least one a week) and weeks (one out of 4) so your body is fresh enough for you to ride hard. Do your hard days when you're freshest (if you took the weekend off, Monday is great). Use a comfortable non-fatiguing cadence that's likely to be somewhere between 80 and 100 RPM. Your problems are not being able to get enough oxygen to your muscles and process lactate fast enough, not strength of which little is required on a bike and trying to push a bigger gear is only going to limit how much you can do.

You don't need to be religious about it. If 2x10-20 minutes is too structured hard efforts on 3-7 mile stretches of road between convenient landmarks will do fine. If you miss a day it's not going to be the end of the world.

That's fine. You'll just be going faster than you would be when working just as hard up-hill which means the cooling breeze will be more refreshing.