Does Edward bite Bella?

No, he doesn't "bite" her, but she does end up becoming a vampire. When she's having Renesmee, she basically dies and he injects his venom into her heart. So Bella had her human death on her daughters birthday.

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Edward first plunged a syringe in her heart with venom after it stopped when Renesmee was delivered. If you continue about half a paragraph down, Jacob describes what happens when Edward is biting her.

"It was like he was kissing her, brushing his lips at her throat, her wrists, and the crease at the inside of her arm. But I could hear the lush tearing of her skin as his teeth bit through, again and again, forcing venom into her system at as many points as possible." {| |- | Edward does not bite Bella, but he changes her into a vampire in Breaking Dawn to save her life. |}