It can take a while. Some people definitely pick it up easier than others. Just keep putting one foot in front of the other. Most people (especially when just starting off) can't add both speed and distance at once. Normally by adding the distance and building the endurance you will naturally start to pick up speed as you progress. I don't remember if you are following a training program, but most likely at this point if you are training for a half, you would want to be slowly adding distance to your runs at a constant pace. Maybe 1/4 to 1/2 mile at the most. And when you do add distance, especially starting out you need to be able to repeat that distance. So if you run 3 miles this week, you want to repeat that next week, and then try for 3 1/4 the week after that. (Not those distance and time spans specifically, but you get the idea)
The weight loss will come over time. Some of it does depend on your heart rate. Yes you burn calories by simply moving, but I believe if you hit a certain heart rate zone you burn them more efficiently (suggested zones are often on the cardio equipment at the gym). And as @
Kristia said, make sure you are doing some basic weights as well. Muscle helps to burn more calories even while at rest.