Target HR and Eating Back?

I had an overuse injury in my knee and through rest and stretching, it's mostly back to being pain-free and functional. I've also been away from home and eating like ... er ... well, not so well. So between the two, I need to get back at it and have downloading myfitnesspal to my phone and gotten back into my running shoes. Soon I'll be home and back in the saddle, (yes, of my very own horse) too. I only have a few pounds that I'm not happy about, but need to eat much better and actually get fit and healthy.

What's the deal with 'target heart rate'? I read that I should be at about 160bpm, but when I do that I don't feel like I am really pushing myself, and I really want to be able to run faster, farther (and I have a goal of being fit enough that my resting heart rate drops back below 70). Is it bad to work to 190, keep it there for 5 minutes or so, and then ease up until I go back down to 160, and then repeat? I feel like I am really getting a workout when I do that, but feel like I'm breaking some 'rule'. And I don't really want to hurt myself again.

I am learning to watch and log what I eat, but on days I exercise and burn more calories, how important is it to eat more calories to compensate and keep my 'net' intake above 1375 (estimated BMR)? Does it really matter all that much on a daily basis, or can I just make sure that the weekly total averages out high enough?

I am 'hungry' about every two hours, but if I'm otherwise occupied, the feeling passes quickly. I do have a problem with eating within an hour of exercise though, either before or after, as it tends to make me feel nauseated. I haven't been 'eating back' because, well, it feels like eating for the sake of eating, which is a habit I am trying to stop... I am open to the idea that that is not actually the case and I should, but need some reason/logic to overcome my (potentially irrational) aversion to the idea.