1. Whitebeard wasn't injured in any way that significantly hindered him. He commented that Squard's stab wasn't enough to hinder him. If you're referring to his illness, then so? If Whitebeard loses to someone because his illness hindered him, then WB<than person due to his illness. Not only that, but Whitebeard's illness was not affecting him at the time Akainu cleaved off half of his face. This is evident by the fact that Whitebeard was striking Akainu, a Logia, meaning he was using Haki, and his illness stopped him from using Haki when he wanted to knock out Ace's executioners with Haoshoku and when he wanted to strike Kizaru, it left him unable to use Busoshoku.1. Exactly my point if Whitebeard wasnt injured before hand Akainu couldve never take off half his face he would push Whitebeard yes only because Whitebeard had heart problems.
2. Whitebeard did school him U_U Whitebeard only punched akainu 3 or 4 times in the fight while akainu was giving it his all and after whitebeard delivered the island busting punch to Akainus ribs he fell down in the ground hurt and in pain shouting "Shirohigeee"!!!! and couldnt fight back while whitebeard was just standing over him while hes on the ground why didnt he get up straight after he had recieved the blow???? exactly schooled. Akainu was initially meant to die at Marineford Oda just gave him plot shield to make things more interesting for the story.
2. No he didn't. Whitebeard snuck up behind him and hit Akainu with a sneak attack. And even then, Akainu was still able to cleave off half of his face. If I snuck up behind you and stabbed you in your kidney, you're gonna faulter too, even if you were better than me. One vs one, face to face, with no sneak attacks, Whitebeard vs Akainu ended up as Whitebeard on his knees, magma in chest as Akainu stood over him. Was Akainu hurt? Yes. But he was not schooled, because he wasn't even defeated. If Akainu was defeated, he would have stayed down for the rest of the war. If Akainu was "schooled", he wouldn't have had enough strength to face off against all of Whitebeard's commanders(minus Jozu), two Shichibukai, and Ivankov. If Akainu was schooled, he wouldn't have been able to keep on fighting and defeat Jinbei with absolutely no difficulty.