I'm going to number your questions, just to make it easier for me to organize my thoughts.
1. Puppy chow: It depends on the breed. I'm assuming the puppy is a beagle, based on your second question? He should be on puppy food until he's about a year old. Large breed dogs need puppy food longer because they continue to grow bigger longer than small/medium breeds do, but the "standard" for non-large breed dogs is about a year. Though I will admit that we switched my corgi to adult food when he was 9 months old. He ate like a bird until he was about 6 months old, then all of a sudden, he was mister hungry mouth. Went from being average weight to slightly overweight very quickly. We wanted to get the extra weight off of him as soon as possible so that he didn't suffer any adverse affects or continue to be overweight. Because puppies grow so much and are generally so active (my corgi was a lazy puppy-part of why he gained weight when he started eating more), puppy food is much higher in protein/etc than adult food. My corgi didn't need the "extra" in the puppy food, so we switched him to adult food. A few months later, we switched him to lite adult food. He's now three years old and average weight.
2. Honestly, I would be very surprised if you could get him to stay in his hard with no type of fencing (electronic or physical). Some dogs are very much homebodies. But beagles are hunting dogs/hounds. They are bred to follow their noses... I honestly wouldn't trust one to stay in the yard without fencing. Too much of a chance that they will get distracted by a squirrel or a scent and take off running. Maybe someone else will have better advice. The only one of our dogs that I will let out of the house without a leash (unless we are just going from the door to the car) is Tucker, my lab. We adopted him at a year and a half old. My mom nicknamed him Old Faithful. We didn't train him to stay in our tiny yard... he just naturally wants to be with his family. So when we go potty at bedtime, he is allowed off the porch without a leash because I know that he will go down the steps, pee in the front yard, and come right back as soon as he is done. It's just his personality.
3. The first thing that came to my mind is that he is acting scared... I wonder if something happened one time that has him scared when he hears the noise of someone walking into the room. I'm not really sure.