The definition of quantities in special relativity as upper-index or lower-index My question is for Minkowski metric $\eta_{\alpha\beta}=\mathrm{diag}(1,-1,-1,-1)$. While defining quantities like the four potential, four momentum or even space-time interval for that matter, why do we prefer defining them in the fashion 
$$A^{\alpha}=(\phi,\vec{A})$$ and not $$A_{\alpha}=(\phi,\vec{A})~?$$
Also, if we define one of the quantities using upper indices, is it mandatory to define all other quantities with upper indices? 
 A: There are two objects in relativity that contain the same information, but are of different mathematical nature - vectors and covectors. Covector is dual vector to the original vector. That is, it is member of different vector space, but (in space endowed with metric) there exists natural  one-to-one correspondence between vectors and covectors. 
Now, since they both convey the same physical message, we want to call it the same name, but still, we need to distinguish them in order to be clear on mathematical formulas - and that is achieved by using different position of the index. So once you define vector with upper index, you want to make all vectors using the same convention to not confuse them with covectors. In STR this is more or less just mathematical curiosity, but in GR with its curved spacetime the notion of vectors and covectors is very useful tool.
A: This is an arbitrary convention that was fixed historically around the time that Einstein published the general theory of relativity. It's similar to the right-hand rule for defining torque, or the convention that the charge of the electron is negative. Although it's arbitrary, it's fixed, so different authors do not use different conventions as a matter of personal choice. (In this respect, it's different from conventions such as the signature of the metric.)
The convention is that displacement vectors are written as upper-index quantities. (In general relativity, we would have to talk about infinitesimal displacements or tangent vectors.) Once this is fixed, we have conventions that then flow from that convention. For example, if we define the velocity vector as the derivative of the position with respect to proper time, then this makes the velocity vector an upper-index vector.
Of course you can lower an index on a velocity vector and make a velocity covector, and that's fine. But the connection between the components of these quantities and physical measurements using clocks and rulers is different for the upper-index quantity than for the lower-index one, and this is not a matter of individual choice, except in the sense that it could be an individual choice to use the symbol 2 to represent the number three.

Also, if we define one of the quantities using upper indices, is it mandatory to define all other quantities with upper indices?

It isn't quite that simple. For example, having fixed displacements to be upper-index, it follows that gradients are lower-index. And not all quantities in relativity are defined by taking nonrelativistic quantities and sticking them together to make four-vectors.
As an example, the stress-energy tensor of a perfect fluid can be defined, in the $+---$ signature, as $T_{\mu\nu}=(\rho+P)v_\mu v_\nu-Pg_{\mu\nu}$, where $v$ is the velocity vector of the fluid's rest frame. This is exactly the same as the definition $T^{\mu\nu}=(\rho+P)v^\mu v^\nu-Pg^{\mu\nu}$, since either of these equations can be found from the other by raising or lowering indices. But, as discussed above, the four-velocity has an interpretation that ties it to the convention that displacements are upper-index, so this has implications for the stress-energy tensor's representation. Now suppose that the metric has the form $g_{\mu\nu}=\operatorname{diag}(A^2,-B^2,\ldots)$. Then the various forms of the stress-energy tensor then look like the following:
\begin{align*}
  T_{00} = A^2\rho   \quad & \quad T_{11} = B^2 P \\
  T^0{}_0 = \rho   \quad & \quad T^1{}_1 = - P \\
  T^{00} = A^{-2}\rho   \quad & \quad T^{11} = B^{-2} P.
\end{align*}
So if you want to connect the components of the stress-energy, in these coordinates, with the physical observables $\rho$ and $P$, it's not arbitrary which forms you consider to represent which physical measurements.
