No, it is not destroyed. If the Z/X that attacked B01-053 "Lawbringer, Batura" is put into the Player Square when the effect of B01-053 "Lawbringer, Batura" resolves, that Z/X is not destroyed.
Yes, it is put as Sleep. If a Z/X is put into another Square due to ability, it is put without changing its Reboot or Sleep state.
B01-053 "Lawbringer, Batura" is attacked by B01-056 "Fierte the Pride". At the end of battle, the attacked B01-053 "Lawbringer, Batura" survive, if B01-056 "Fierte the Pride" is put into the Player Square by effect, does B01-056 "Fierte the Pride" is destroyed?
At the end of battle that B01-056 "Fierte the Pride" attacked, when it is put into the Player Square by its effect, does B01-056 "Fierte the Pride" is put as Sleep?