There are a lot of red flags here which make me think this is a scam... I strongly suspect OP has been scammed and there are no coins to recover, but we can at least try.
you may be right. the only thing is this was back in like 2010. i'm not sure how people sold bitcoin back then. maybe it was accepted to just sell the private key and not actually send bitcoin to the person's address. obviously that seems like a really bad idea today and you would think it would seem like one to people back then too but they didn't have all the different wallets and things they do now. so i guess it was harder to create your own wallet/address. there is also the remote possibility that the OPs laptop got some data corruption and the private key is not correct due to that. that's probably very unlikely but i guess it is a remote possibility.Quote
You can correct me if I'm wrong (since I've never used PayPal), but my understanding is that there is a time limit of 180 days to open a dispute. PayPal are not going to entertain a dispute for a transaction which is in the region of 10 years old.
Well of course they're not going to let someone dispute something that old. I didn't realize this at first but it seems the OP probably would not even have been able to dispute it 10 years ago either since back then I don't think paypal had a digital goods protection. Now they seem to though: https://newsroom.paypal-corp.com/news?item=122644 so that would have been another reason to not be using paypal back then to buy btc. scammer's paradise... 
no one should be needing to buy bitcoin from someone else using paypal these days though. since they can just buy it from paypal directly. the irony...