No, this is not possible. PayPal only supports:
transfers between two PayPal accounts, and
transfers between a PayPal account and its associated bank accounts or cards.
This makes PayPal unsuitable as a general payment method or as a general-purpose bank account.
To transfer money to a recipient who doesn't want to open a PayPal account, you will need to ...
The emails are just notices. You may delete them if you think they're suspicious.
The fact is that IF the money came into your account someone sent it to you.
Could have been a mistake.
You can start a case with PayPal to see if they have any further information.
In any case, I would suggest you to just leave it there for 180 days. That's the maximum ...