I think it is wonderful to have goals and dreams, wherever you are in the process to achieving them. I do think that worrying about the logistics about it at this point is definitely a little ahead of yourself. But if you want to daydream and make sure that you are developing helpful skills for when you are ready to apply- awesome. I think the youngest person in our group of 87 volunteers was 20... so, a few more years than you
To answer your logistical questions- every country approaches the differently, to ensure volunteer safety and be respectful of the host culture. Some countries set you up with an apartment when you move to your permanent site after training, with the help of your counterpart. Others make it mandatory you live with a host family throughout your permanent placement. Others recommend a host family but allow you to find your own apartment after you have been there a certain amount of time, if you want. Your rent, whether it is with a host family or on your own, is sometimes paid directly by your host organization. If they cannot afford it, the PC gives you a separate allowance for rent, electricity, and water, based on the exact cost of your bills.
In most countries, the monthly allowence is deposited into a bank account the PC will set up for you. Depending on what your location is like, you may have to withdraw it all at once or you may just keep it in your account if you have a bank nearby. It is supposed to be the average pay that the people you are working with would make, but it is adjusted based on a cost of living survey completed by volunteers every year. It allows you to live rather comfortably if you adapt to the host country standards of living with occasional splurges. Volunteers that cook and eat like they would in the US, purchase new, brand name clothing, travel and party often find themselves out of money by the end of the month. Any travel for PC related work, such as conferences or helping another volunteer on an approved project, is covered by the PC, along with a rather generous per diem for travel days. Healthcare is also covered. So your allowance only has to go to your own food, extra supplies for work, clothing, hygiene, and entertainment.
You wont be able to take a pet with you from the US. At least, I have never heard of that. If you are, they would have to be in quarantine in your host country for a significant time, which can be stressful to pets. Quite a few volunteers adopt pets while they are in their host country, but then there is the issue of either finding someone when you go home to take care of them, or paying to get them back to the states, and, again, making them go through a long quarantine.
One thing you might want to check out now to prepare yourself for PC (it gives you great skills and looks awesome on the application) is AmeriCorps. They have short programs (from a month to three, usually during the summer) and year-long programs if you are considering a gap year before university. They also give a small bonus to help cover the cost of continuing education.