And can this be used on an account which has had a Prime membership in the past from the student email ?
Yes, you can add it to any Amazon account you want but it won't extend what you already have only replace it. He's just adding you on HIS account (or someone's account he has) as a "family member." When you have Prime you can add 4 family members at no charge. All they get is the free 2 day shipping and no other prime features. If he sells 4 family members at $25 each his $99 PRIME with all the features is free.
Nothing wrong with doing that as long as you know the limitations. You just need to make sure the seller doesn't disappear, drop you off his PRIME then go and resell it again elsewhere but from his "trust" he seems quite safe.
It does have trust but again the password was changed recently , so could be risky as his account wouldn't be worth the amount. Can someone with a family account remove another account he added as family from Prime ?
Why does changing password look bad? Aren't you supposed to change your passwords regularly?
As for your question, yes. I've been doing the same thing (not on this forum) for over 5 years with my account and never kicked anyone off.