Yes, theoretically you can, although you will be restricted in the number of card payments that you can accept. Realistically you need to upgrade to a Premier account to use eBay.
All you have to do is sign into eBay and next to the item that you would like to pay for click on make payment and choose PayPal as your payment option. You will be re-directed to PayPal and asked to...
No, you don't have to have a PayPal account to make a Paypal payment. You can make one-off payments through PayPal using your credit card, without registering.
It really depends on what the seller wants. Sometimes he/she will want you to pay with PayPal ONLY. But, maybe they want cash only. Or, they don't care.