I'm no expert, but I might be able to offer something regarding your breathing. Depending on exactly what you mean by 'panting', it's generally not great unless you're really exerting yourself.
Everyone's different, but a lot of people find themselves on a routine, gentle run (so on smoothish, flat terrain) settling into a 1:2 ratio - one breath in for every two steps, then one breath out over every other two steps.
I might be corrected on this one, but if you find yourself panting or really struggling for breath, then listen to your body and slow down until your breathing comes under control. Exceptions to this might be sprints, or times when you're really trying to push yourself for a certain goal, but otherwise chill out until you can breathe comfortably and you'll run for a lot longer.
I agree with the breathing advice. Regarding the shin splints, these are typically a sign of calf weakness. If you are new to running or have taken a significant period of time off from consistent running, then you should ease back into it. Shin splints typically become a problem when you are doing too much too fast. I understand the pressure to get fit by the start of the season, but running too fast too often won't make up for lost time and is more likely to set you back.
Sounds to me like the shin splints are related to your stride, effort and cadence. During a very EASY, try shortening your stride, striking the ground under you and increasing your cadence a little in general.
To understand foot strike placement, simply run in place for 30 seconds and see where you naturally strike the ground and then try too mimic this placement while moving forward.
Yeah. If you are getting enough rest, then it shouldn't be a chronic problem. Looks like Eric has some good advice for you below. Good luck, Daniel.