Imported video getting out-of-sync audio in Adobe Premiere
It sounds like your source media went through quite a lot of different operations, which makes it difficult to say what exactly has upset the audio/video sync. 

I would start by trying to isolate the problem: repeat the process on a smaller scale (just one cut/edit, instead of the whole thing) and verify the output at each stage. That will help you understand where exactly things have gone wrong. Then, you could edit your question and present us with an isolated example of an operation that affects the sync. That should make it easy to resolve the problem.