What if in magic the gathering a double strike creature attacks a first strike creature?

Well, before I say anything else, remember that creatures can never attack other creatures directly. You can only attack your opponent (or a planeswalker they control), and then they decide how to block your creatures with their own.

Say you attack your opponent with Student of Warfare, and he blocks with his Baneslayer. In the first strike damage step, the Student deals 4 to the Baneslayer, and the Baneslayer deals 5 to the Student. State-based actions see the 4/4 Student with 5 damage, and kill it. Since it's not around during the regular damage step, it can't deal any damage from its second strike.

Yea. Also if a creature has double strike and trample, if the first atk would kill the creature, the second would do its full atk directly because it's cosidered to be blocked by a 0

Think of doublestrike as first strike + normal attack. So, when your creature is blocked by another creature with first strike, they both deal damage to each other. Your creature would have to live to continue fighting.