When I came to the US, I had an H1-B visa. I went through all the paperwork and after some time recieved a Permanent Resident (Green) Card for myself and for my wife. At the same time I also recieved an Employment Authorization Card (EAD). The green card lasts 10 years but the EAD only lasts 1 year. Do I need to renew the EAD since I am a permanent resident?
You do not need to renew your Employment Authorization Card. Your Green Card is sufficient proof of your ability to work legally. If you plan on applying for US Citizenship (which you should), please educate yourself on the residency rules if you spend much time abroad. They are the reason most often why people cannot apply.
I am not sure how you obtained your Green Card (family or employment based), but assuming it was lawfully obtained, your Green Card is sufficient for employment, admission, and status purposes. A adjudicative officer who interviews an applicant will often take the EAD from the applicant once they determine they are going to approve the case. If you still have the EAD, I advise putting it in a safe place, or destroying it so another person does not impersonate you.