1) I want to increase my speed over distances and climb hills faster with less fatigue.
You need to increase both aerobic capacity and strength. It takes strength to climb and ride fast, but you have to have the aerobic capacity to keep it up for more than five minutes.
You're not primarily interested in fat burning, but a fat-burning metabolism is a side effect of a good aerobic base. As Modo notes, your caloric output is related to your level of effort, not to your heart rate, so don't think in those terms.
For strength you want to drive your muscles to exhaustion, then let them recover. Feed them protein and they will grow. In the context of a ride, do intervals of maximum effort. Your heart rate will spike since you're driving the muscles into oxygen debt. After the interval let your heart rate come back down under 80% (The aerobic range is roughly 65-80% MHR, and since the upper limit is defined as the rate at which you can supply O2 to your muscles, it will also be the point at which you recover your breath.) Do this several time during the ride. Then take a day off or use other muscle groups. (With your schedule that doesn't look like a problem.)
You don't want to ride your whole ride with your HR above the "aerobic" range. That means that your limiting factor is your base, not muscle exhaustion, and you'll have a hard time building strength. One of the best measures of aerobic fitness is how quickly your heart rate drops when you back off an interval.
If you need to develop a better cardio-vascular (aerobic) base, then riding in the aerobic range (below 80%) is the more efficient way to develop it.
Given that we don't know where you need the work, Modo's advice about mixing it up is about the best we can do.