I know that for the Honor Code I have to not post any of my solution code publicly. Is the thing different for the final project? The final project has to be different between user, so I don't think to help anyone by publishing my code.
And what if my code includes some distributed code of pset7/pset8? Is there any kind of copyright problem?
That's not a problem at all. It's your project written by you. One of the main points of final projects is that you'll be able to use them to demonstrate your skills, which means you'll likely be including the project on your resume.
Hosting your project on GitHub is fine, and I'd actually encourage it. Learn to use version control and get involved in the GitHub community.
I also don't believe there's any issue with using some of the code from the PSETs.
If the code is written by You, you are absolutely free to use it anywhere you like.
And for the final project, You want to post on GitHub, I would suggest to upgrade your repository to private so that the final project don't displayed publicly. GitHub provides students a 100% discount on their upgraded membership plus Provide you with many of their tools for free, which could be expensive for you otherwise.