Dude, I think it's excellent that you are thinking so far ahead. These are things that many of us wish we had done back when we finished our classes. Plus, it will relieve the stress of having to get your LORs together when your time comes to apply. However, there is one downside to getting a letter this early and that is if you see yourself having a future contact with this professor (maybe working in his lab for example), in which case it could be better to have him write the letter for you closer to application time. Otherwise, just ask him to write the letter now and have him send it to your schools career center (you would need to open up an account with them first) or if your school does not have a career center which gathers LORs, set up an account with Interfolio. And for your last question: Yes, it is permitted. Good luck!