Absolutely open a separate bank account. If you give them your regular checking account they can lock it down if there is any problem...
There is a lot of information of questionable veracity in this thread...
1. PayPal cannot and will not withdraw anything from your account without your express approval. Period. This has gone to court (years ago) and been resolved.
2. PayPal can't lock down your checking account. No way, no how. They can lock your PayPal account, but that's a given.
3. A separate account for PayPal does nothing. See #1 above. Were #1 not true regardless of what was in the account you'd just end up with an overdraft (with the attendant fees and interest to boot.)
4. We've been using PayPal for many years (almost since they started.) We've had a couple of claims. PayPal locks the amount of the claim (again, another court case established the rules for this) and no more. If there is no money in your account, then they either lock when more money in your PayPal account becomes available or they go to collection (only if they find for the other party.) PayPal has resolved ALL of our cases in our favor. The longest we've had to wait for a payment was about a week or so (buyer claimed they'd been hacked, but since we shipped to their address PayPal said: "bogus" and gave us the money.)
You DO NOT need a PayPal account to buy on eBay. You can pay with a credit card (still processed by PayPal but that's not an issue.) If you sell on eBay you must either have a PayPal account, an account with an approved electronic payments processor, or a merchant credit card account with proper online payments processing.
There have been few problems with PayPal (or eBay) over the last ten years. They both have cleaned up their acts greatly (mostly because the courts forced them to) and the few you see today are often times just fraud cases where the scammer things or hopes to get his way.