I'll give it a crack for you. Can he cross is not a question you should be asking cause he's not really a winger, he's an attacking mid. A proper attacking mid, not a playmaking one. He's not going to contribute too much to our pattern of play, or the control of games that we have but he will be trying to win us matches. He's not going to give us much width either, he's going to give us central penetration. He'll be going where the space is, not hanging around the touchline.
He's fast, real fast. With or without the ball, it doesn't actually make too much of a difference. He's these players that glide with the ball at speed(think kaka) especially in transition when There's open spaces. In those situations he'll beat men easily, in some cases more than one at a time. Put him in a static one on one and he's taking us nowhere.
What makes him a great player? His movement in the final third. He's one of the best off the ball movers in the game today(alongside Ronaldo and Hernandez IMO). You'll notice in situations where There's space, he'll be there in a jiffy ready to finish. If There's a gap between the midfielders and defenders, he'll be there in a jiffy ready to carry the ball. He's a very intelligent footballer and without being a remarkable passer, he creates lots of chances cause of that.
His biggest weakness? That's his finishing. Because of how well he moves, he finds himself in great positions lots of times in games and he has a tendency of not scoring as often as he should. His ball striking technique is nothing special and he'll certainly miscue a few. In Klopps season he really struggled to find the net despite numerous opportunities, let's hope he comes with his scoring boots.
How good is he with the ball? He's very good without being amazing. He's not going to do things with the ball that'll leave you flabbergasted(he's no hazard), but he's got good control and loves to turn with the ball. Technically, he's about in the Lallana category.
Hope that answers you question. In a nutshell, he's a very effective double footed attacking mid with pace, football intelligence and phenomenal movement. What has me very excited about this, and something that makes me believe he'll certainly be better than KDB and Ozil next season is that he fits the way Jose plays like a glove. I'm expecting nothing less than Mahrez numbers from him, nothing less.