Inputting a superposition into a cNOT gate
This is more of a question on linear algebra rather than a question about quantum computation. You should probably have a firm grip on basic linear algebra (vector spaces, linear operators, inner product, etc) before delving into quantum computation (there's no way around it).

CNOT is a linear operator in our vector space, $\mathbb{C}^4$ (were only dealing with linear transformations, specifically unitary). In order to specify how a linear transformation operates on an arbitrary vector, it is enough to specify how it operates on each member of a basis for our vector space. Suppose we apply some linear operator $T$ on an arbitrary superposition, then by linearity:

$T\left(\alpha |00\rangle + \beta|01\rangle+\gamma|10\rangle+\delta|11\rangle\right)=
\alpha T|00\rangle+\beta T|01\rangle + \gamma T|10\rangle + \delta T|11\rangle$

$T|ij\rangle$ is the application of the operator $T$ on the vector $|ij\rangle$. The application of a linear operator in a finite dimensional vector space can be described by matrix multiplication, so this means multiplying the matrix representing $T$ by the vector $|ij\rangle$. The "order" you refer to does not matter, because addition in a vector space is commutative.

Since you  know how to apply CNOT to the four basic states, the above equality shows you how to apply it on an arbitrary superposition. 

When talking about matrix representation, the vector $\alpha |00\rangle + \beta|01\rangle+\gamma|10\rangle+\delta|11\rangle$ is represented by 
$
\begin{pmatrix}
\alpha\\
\beta\\
\gamma\\
\delta
\end{pmatrix}
\in\mathbb{C}^4$, so applying CNOT to this state means multiplying it's matrix by this vector, i.e.

$CNOT\left(\alpha |00\rangle + \beta|01\rangle+\gamma|10\rangle+\delta|11\rangle\right)=
\begin{pmatrix}
1 &amp; 0 &amp; 0 &amp; 0\\ 
0 &amp; 1 &amp; 0 &amp; 0\\
0 &amp; 0 &amp; 0 &amp; 1\\
0 &amp; 0 &amp; 1 &amp; 0 
\end{pmatrix}
\begin{pmatrix}
\alpha\\
\beta\\
\gamma\\
\delta
\end{pmatrix}=
\begin{pmatrix}
\alpha\\
\beta\\
\delta\\
\gamma
\end{pmatrix}
$.

This is obviously equivalent to applying CNOT to each of the four vectors in $\mathbb{C}^4$ and adding up the results.