Sure. Let me go down the list.
: Akali doesn't have quite the burst of other assassins when not fed, and her smoke cloud costs a lot of energy. If you use your abilities on her face right after she R's you then her smoke cloud channel won't have enough time to go off to dodge your skills. In lane (I assume top specifically? Since her heal is hard to get off early in mid.) early just try to stop her from farming to delay her hextech gunblade until it is less relevant. Don't worry as much about killing her. If she starts to snowball you will have to play safe until you are clear to shove lane and teleport bot or go for an objective.
: As melee this one is rough. Watch the timing on his Q and try to get on top of him right after he uses it since he can't reposition his minions for his jump while it is on cooldown. Basically, avoid the minions he spawns and he has no dps. If he ults defensively just peel off unless you have an easy way around it.
: Like I said, a little reduced early game damage would be fine here, in my opinion he is a little strong. When fighting him attempt to CC him out of catching his axes, since that is where all of his damage comes from. If you can cause him to drop an axe or two you will be quite a bit better off, or try to start a fight after he misses one while farming, but that won't happen often. Another thing you can do is aim your skillshots where his axe is going to land so he either has to tank it or miss the axe.
: Quicksilver sash for his ult. Kill the little minions when he spawns them. Dodge the Q. Don't stand near minions with aids when they are about to die. Also try to poke the bubbles off of him before you engage if you can. Most skillshots outrange everything he has but the Q.
: Him using his ult is a very good indication of what he is going to do next, and you need to have patience. He is either going to run strait at you and attempt to alpha strike you after you stop to cast a spell, or he is going to take a longer route to the side of you and attempt to circle you to make it hard to hit him.
If he is running strait for you then you have a couple options. You can either wait until he is so close that he can't alpha strike out of cc due to the limits of human reflexes and the inherent delays of online games, or you can try to bait the alpha strike by casting a non cc spell or autoattacking and then CC'ing him after it is done.
If he is circling you it is a little trickier, but basically just wait until he is so close that you cant miss and he cant react.
Also, grievous wounds helps here. If he is getting super fed thornmail will lower his dps on you a lot. If you already have grievous wounds in another items I really like having randuin's omen vs him though, the active doesn't slow him during his ult but the lower crit damage helps a lot. If you still need more damage and can't invest that much gold into one of those items, then even just a warden's mail or bramble vest help a ton and are only 1k gold.
Even with his ult it is very difficult for him to get on an important target and stick to it. Just kite him until it is over. Remember, if he has to move between attacks it will drastically lower his dps. Also, if you can try to lead him away from his axes so he cant lower its cc. If he stops to attack a tank or something then turn on him. And don't forget to get grievous wounds against him to help against his inherent lifesteal.
When he is powerballing towards you remember that he can't go through your minions so putting some between you and him can be the difference between getting caught or not. Don't panic too hard when he catches you though, he can hold you down for a long time but he doesn't do much damage by himself, so you will probably get a chance to run after he catches you if his team can't catch up. In lane his taunt won't be very long since it raises with level and I believe he maxes W for fighting or Q first for ganks. Been a while since I brushed up on his skill leveling order so the thing to do might be taunt first but I doubt it.
She is a late game hypercarry, avoid letting her get fed early. If you only fight when you are sure to win and back off when she ults you should be able to outdo her late game with fairly even farm if you don't let her get fed. When you are fighting her stay away from walls and save your skillshots until just after she uses Q so she can't use it to dodge. Her Q is a big part of her dps so if she never uses it then you should be coming on top in dps.