Your actual question has nothing to do with the technical issue of linking a PayPal account to a bank account. It is all about the accounting of the money. That is, what you claim as income and what you can prove to the taxman.
Yes, you will need to separate the money. Linking to a business account is probably the way to go. From there, it is about how you keep track of the money and account for it.
How you do the accounting is a different question.
No, it does not automatically become business income just because it goes into a business bank account. You still have to keep track of said income and claim it somewhere on some tax form(s). The point of the separate business account is to avoid the commingling of the the money which may lead to you losing the liability protection of an incorporation. The bank doesn't file your taxes for you.