Are component of vectors vector and can we divide them into components? We know that two vectors can sum up and make one vector. And dividing a vector into components is the opposite process of summing vectors. Then the components of vectors should also be vectors. Are they really vectors? If they aren't, why are they scalar?
And if they are vectors, can we divide these components into more components? 
 A: 
And dividing a vector into components is the opposite process of summing two vectors.

It's not the "opposite process" because, given two vectors, there is only one sum; but given one vector, there are infinitely many pairs of vectors that sum to the given.
When you decompose a vector into components, you do so with reference to a set of basis vectors, and the components are scalars, by definition.
Adding vectors, on the other hand, is a "pure" operation that can be defined without talking about components or any coordinate system.
A: Often when we speak of the components of a vector, $\vec{a}$, we mean  vectors ($\vec{a_{x}}, \vec{a_{y}}, \vec{a_{z}}$, say) in particular directions  that add up vectorially to make the original vector, $\vec{a}$.
Sometimes, though, we mean ($a_{x}, a_{y}, a_{z}$): the scalar coefficients of the unit vectors in those directions. So…$$\vec{a_{x}}=a_{x}\vec{e_x}\ \ \text{etc}$$
Here's an example where we use scalar components. The cross product of two vectors is given by$$
a⃗ ×b⃗ =i⃗ (a_{y} b_{z}−a_{z} b_{y})+j⃗ (a_{z}b_{x}−a_{x} b_{z})+k⃗ (a_{x} b_{y}−a_{y} b_{x})$$ 
The safest thing is to follow Synge and others and to talk about either 'scalar components' or 'vector components' depending on what we mean! But most of us just refer to 'components' because it's almost always clear from the context whether we mean vector or scalar components. 
You can further divide vector components into vectors in different directions, because vector components are, well, vectors.
You can't do this with scalar components, though this never causes trouble, because people revert (probably without thinking) to regarding components as vectors when they want to resolve again! 
A: I think we can divide vector components into more components in a sense that, if done correctly, this is not going to violate any physics laws or lead to wrong results.
For instance, we can say that, after breaking vector components to sub-components any number of times, the vector sum of all the resulting sub-components will still be equal to the original vector.
I think we can also say that a vector could always be broken, directly (i.e., without going through Cartesian coordinates first) to any number of random components, given that all redundant components are pre-assigned some values. 
For instance, in a 2d space, we can randomly choose n unit vectors (where n≥2), assign random values to all but 2 of them and then determine the component values for the remaining two.
So, if someone finds it helpful to break vector components to any number of sub-components or break a vector directly to any number of components to start with, there seems to be no cause for concern.
