Paypal is actually a separate company although it was started by Ebay. That is why they want a username and password. It is a separate account from your Ebay Account. I would suggest using a different password but use the same User name.
I've never had any problems with it. When you buy something with Paypal, the money is withdrawn from your account by EFT or you can specify a credit card. The default however is your bank account. If you sell something you have to tell them to transfer the money into your account.
As for the seller's fees, this is how Ebay makes it's money. It's just like selling something in a consignment store. They handle the logistics of the sale and keep a percentage of it.
It's ok - not shady at all. I use ebay all the time and have never had a problem. I only use paypal for payment. In fact, I use my paypal account for all Internet based purchases or sells, whether it's through ebay or not. Paypal is perfectly safe.
Ebay doesn't ask you for your bank account details, maybe bank account number and holder, but never login information for your online banking account, that would be a scam site. You can pay your ebay fees using a credit card.