How long did you live in Canada? There is a residency obligation to keep your PR and it involves spending at least 2 years in Canada in any 5 year period. Since you have been a PR for more than 5 years, they will look at the past 5. If you meet this residency obligation, that is, you spent at least 730 days in Canada in the past 5 years, you can apply for a travel document from an Canadian embassy in the US and when you get it, go to Canada and apply to renew your PR card.
However, if you do not meet the residency obligation, you are really not in the position to demand anything from immigration. If you want to attempt to save your PR, you can tie up all your loose ends in the US, go to the border and state that you are a PR but you have lost your documents. If you have your landing documents, you can use them as proof of your status. Otherwise, they might refuse to believe that you are really a PR and send you back. If they are convinced that you are a PR, they will ask you some questions like how much time did you spend in Canada in the past 5 years, do you meet the residency obligation and if not, then why not, when did you lose your documents and did you report your lost documents to immigration or the police. If they are convinced that you are a PR, they have to let you enter but they could report you for not meeting the residency obligation. If you want to keep your PR, you would have to appeal for it within 30 days of having been reported. You would be allowed to stay in Canada as a PR during the time it takes to process your appeal which could be 1-2 years.
Winning your appeal would be based on the reasons why you did not meet the RO. They may give you a chance if you had H&C grounds like medical reasons, taking care of a sick family member etc. Choosing to live outside Canada and studying outside Canada would not be seen as a good reason.
