As others have said, 13 miles is decent, and nothing to be ashamed of. My old commute was 9 miles round trip, and I normally carried a backpack, in part because I didn't trust my laptop to a rack bag. I do have a rack, and occasionally used it to augment. You will be able to find seat-post mounted racks, usually with a load limit of around 20 pounds, that will work fine with your full-suspension bike.

As for tires, that kenda does look about right. However, I recommend finding a local bike shop, for two important reasons. The first is that they may be able to teach you some basic repair, like changing a tire, and may have a beginner class to cover such things. The second is that you will end up needing repairs that are beyond your ability, and it's good to have a working relationship built before you bring the bike in the first time.