What is the difference between a chinaman bowler and a leg-spinner?
The trajectories and direction of both the deliveries are the same.
And why do they call it Left arm unorthodox spin? I can understand unorthodox but why just Left hand?
We all know Brad Hogg as a leg spinner. But Wikipedia states him as a Chinaman Bowler.
What is the difference between a chinaman bowler and a leg-spinner?
The trajectories and direction of both the deliveries are the same.
The trajectories and direction of both deliveries are not same.
A leg spinner bowls right-arm with a wrist spin action, causing the ball to spin from right to left in the cricket pitch, at the point of delivery.
While Left-arm unorthodox spin bowlers (i.e. chinaman) use a wrist hand action to spin the ball which turns from off to leg side (i.e. left to right) of the cricket pitch.
Chinaman's delivery vs Leg spinner's delivery
vs 
And why do they call it Left arm unorthodox spin? I can understand unorthodox but why just Left hand?
That is because very rare bowlers bowls this type of delivery. And this type of delivery first seen in 1933 which was bowled by Elliss "Puss" Achong who was a left-arm orthodox spinner. Since the ball he bowled was not orthodox for a left-arm spinner, it was called as unorthodox. The story behind why such bowlers are called as chinaman is given in Wikipedia (emphasis mine).
The name has its origins in a Test match played between the West Indies and England at Old Trafford, Manchester, in the year 1933. Elliss "Puss" Achong, a player of Chinese origin, was a left-arm orthodox spinner, playing for the West Indies at the time. According to folklore, Achong is said to have had Walter Robins stumped off a surprise delivery that spun into the right-hander from outside the off stump. As he walked back to the pavilion, Robins said to the umpire, "fancy being done by a bloody Chinaman!", leading to the popularity of the term in England, and subsequently, in the rest of the world.
We all know Brad Hogg as a leg spinner. But Wikipedia states him as a Chinaman Bowler.
No. Brad Hogg is not a leg spinner. He is a chinaman(or Left are unorthodox spin) bowler as I have already described.
I'll try to clear this left handed spin thing up...
Best to refer to spin bowlers as finger spinners or wrist spinners (even though it is very difficult to bowl any spin ball without both the wrist and fingers co-operating to spin the ball).
A left arm orthodox finger spinner turns the ball, to a right handed batsman, from leg to off, bowling the ball with the front of the hand (mainly) facing the batsman (or fine-leg), and flicking the index and middle fingers and wrist in an anti-clockwise direction (from the viewpoint of the bowler). Can be known as a left handed leg spinner.
A left arm unorthodox spinner (or Chinaman) turns the ball, to a right handed batsman from off to leg, using a similar finger and wrist action but bowling the ball with the back of the hand (mainly) facing the batsman. this is achieved by adopting a wrist starting position rotated anti-clockwise from the orthodox position and flicking the fingers to release the ball. Difficult to master but highly effective. But it is not to be confused with the Doosra which is the same shape of delivery but with the hand rotated clockwise in the pre-release position and delivered with a slight straightening of the arm action (impossible I have found!)
A left handed wrist spinner uses a more wristy action and spins the ball, to a right handed batsman, from off to leg, bowling the ball with the front of the hand (largely) facing the batsman (or third-man) and using the thumb as well as fingers and wrist to impart spin. The opposite of this ball is the Googly which uses the same wrist and thumb/finger action but is delivered with the back of the hand (largely) facing the batsman, which has been rotated anti-clockwise in the pre-release position.
hope this helps.
