Ezreal is kind of a carry. He could fit in the bill of a soft carry but he can't carry as well as true ranged carries. Ezreal should take solo mid unless you can't help it. He needs the level advantage and needs a lot of gold to remain effective into late game. Not to mention, you can snipe people off from mid lane and there is no pressure for you to look after another human being. It's definitely the most ideal lane for Ezreal because Ezreal's no babysitter and can handle solo lane effectively.
If you are forced into a sidelane, try your best to balance last hitting and aggressively harrassing especially if you are up against a squishy. Some laning mates will think ramboing the enemies is effective, thinking you'll be right behind them. I just let them die unless I think there is a chance we could kill the enemy. If you blindly follow your teammate just because he jumped in, it might end up being a double kill for the enemy. Always make sure both of you are communicating and coordinating proper attack.
The worse kind is the kind that complains that you are hogging up their farm. Well, herpy derping do, that's what happens when you lane with a champion who belongs in solo mid. Now, you shouldn't literally steal their creeps (You can tell which one they want because they'd be circling around it waiting for it to go low on health). Just make some kind of agreement with them. You could only go after caster minions and let your partner get the melee. Either way, try to avoid side lane altogether unless your partner is a babysitter like Soraka or Janna.
AD Ezreal is simply better because his autoattacks become strong and he can make use of his passive well with it. With the recent change to his ultimate, there is no reason to go with AP. AP Ezreal can get powerful with the right items but he's an even worse farmer than AD Ez so he takes a looong time to get strong. Same problem with hybrid, except hybrid really doesn't do anything well.