No, that's not weird at all! I've been running with my dog since I was 8 years old and I love it! I definitely feel safer with a big dog, but be sure you're actually safe, PLEASE also carry mace. You never know. No matter where you live...I live in a safe suburb, and the ONE time I didn't take my dog, I got followed! I bought mace that same day.
Don't feel like an idiot just because you think no one else exercises. I didn't think anyone in my neighborhood ran until I started running at 5:00am. That's a popular time for people who work 9-5 because it's cooler than after dinner. Plus, some people just prefer to run inside on a treadmill, or go to a gym or a track.
15 is a great time to start running if you like it. Running for me is not just exercise it's a lifestyle. It's relaxing! I got a running GPS watch so I can explore all over the place and know how far I went. I run lots of road races now and I'm training for a half-marathon in September. I don't know what I'd do without my dog! She loves it too...I've even taken her to a couple 5k races and she did great.
Don't worry about what anyone in your neighborhood thinks, but honestly, it's not weird at all. It's great mental and physical exercise. Just play it safe and enjoy yourself!

Good luck
*EDIT* I also bought a digital camera case that is just a little zipper pouch. It has a loop I can attach to my dog's collar. I put my phone, keys, and a doggie bag in there. It's perfect! My mace clips into my shorts like a pen. It's easier than carrying it all.