Question: > 
> $\mathbf{a}\times \mathbf{b}$ follows the right hand rule? Why not left hand rule? Why is it $a b \sin (x)$ times the perpendicular vector? Why is $\sin (x)$ used with the vectors but $\cos(x)$ is a scalar product?
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So why is cross product defined in the way that it is?
I am mainly interested in the right hand rule defintion too as it is out of reach?

Answer: 
Well, one can see the rule as is in order to agree with the standard orientation of $\mathbb{R}^3$ $(e\_1 \times e\_2=e\_3)$. Or just convention.

There is a way to see how this can come up "naturally", though. 

Given $a,b \in \mathbb{R}^3$, $a \times b$ is the vector which comes out from Riesz representation theorem (a fancy way to say that $V \simeq V^\*$ using the isomorphism given by the inner product) applied to the linear functional

$$\det(\cdot, a,b).$$

Computing the vector yields both the fact that it has its norm being what it should be and its direction also being what it should be.

To get to the other direction, we would have to invert $a,b$, considering $a \times b$ to come from the vector corresponding to $\det(\cdot, b,a)$, an "unnatural" thing.
The real reason is because it is convention to use a right-handed coordinate system. If you switched to a left-handed coordinate system for some reason (I'm not sure why), then you would need to do your cross-products using a left-hand rule to get the same answers. That's really as simple as it gets. It's a result of our conventions in defining our coordinates the way we do.