You need to be much more careful what the "upper" and "lower" indices denote and where they originate from. I'll discuss the two different "types" of upper/lower indices you are talking about:

# Tensor indices

The first source of "objects with indices" is *differential geometry*. On any coordinate patch $U\subset M$ of a manifold $M$ with coordinates $q : U \to \mathbb{R}^n$, the *coordinates themselves* are traditionally written with "upper" indices $q^i$. On the manifold, there are now two closely related, yet *different* objects we naturally want to consider: [Vector fields](https://en.wikipedia.org/wiki/Tangent_bundle) and [differential forms](https://en.wikipedia.org/wiki/Differential_form). One way to define the *tangent space* at a point $q_0\in q(U)$ (corresponding to a point $p\in U$ as $q(p) = q_0$ is as the vector space spanned by the derivatives $\partial_i := \frac{\partial}{\partial q^i}\rvert_{q = q_0}$, whose indices are traditionally placed below. The cotangent space is the dual vector space spanned by the dual basis $\mathrm{d}q^i$ defined by $\mathrm{d}q^i (\partial_j) = \delta^i_j$.

Now, given any vector field $V$, we can expand it in the basis as $V = v^i(q)\partial_i$ for functions $v^i$, where the summation convention is in effect, i.e. we sum over all possible values of $i$. It is the $v^i(q)$ which is what a physicist refers to as "vector". Under a coordinate change, these components transform by the Jacobian matrix of the coordinate transformation. Conversely, we can expand a differential form as $\omega = \omega_i(q) \mathrm{d}q^i$, and it is the $\omega^i$ which the physicist usually calls "the form". These transforms by the inverse Jacobian matrix. vectors and covectors, and likewise differential forms and vector fields are, a priori, *completely different things* and should be conceived of as distinct geometrical concepts.

However, the waters are muddled because in physics we are often on a (pseudo-)Riemannian manifold with a metric tensor $g$ that defines the so-called [musical isomorphisms](https://en.wikipedia.org/wiki/Musical_isomorphism) between vectors and covectors by associating the 1-form $g(v,-)$ to a vector field $v$. Once in this setting, we can freely change the type of tensors and the originally distinct concepts become fully equivalent and interchangable in practical computations.

At this point, I would like to take issue with a certain part of the question:
> Two kinds of indices, covariant and contravariant, are introduced in special relativity. This is as far as I understand, is solely for mathematical luxury i.e., write expressions in a concise, self-explanatory notation. For example, instead of writing the metric as $(Δs)^2=c^2(Δt)^2−(Δr)^2$ one can write $x^μx_μ$ which is not only a compact notation but also tells us that this expression is Lorentz invariant. But both $x^μ$ and $x_μ$, represent same objects: a set of four co-ordinates $(ct,x,y,z)$. 

Although very close to the usage in practice, this is formally just non-sensical, precisely because the geometric objects are not considered properly. If $x^\mu$ is a *set of coordinates*, then there is no such thing as $x_\mu$ - you **cannot** lower the index of a coordinate because it is not a vector or tensor field, and therefore the musical isomorphism is not defined on it. The metric tensor encoded in $\mathrm{d} s^2$ (or $\Delta s$, as the question writes) *does not act on coordinates*, it acts on tangent vectors. The "distance" between two points is given by the extremum of the functional 
$$ L[\gamma] = \int_\gamma \sqrt{g(\dot{\gamma},\dot{\gamma})}\mathrm{d}\tau$$
for paths $\gamma$ between the two points. Since the shortest lines, i.e. geodesics, in Minkowski space are straight lines, it so happens that in *this special case* the expression for the distance between the coordinate points $x^\mu$ and $0$ is given by acting as if $x^\mu$ is a vector and computing its norm with the metric tensor given by the $\mathrm{d}s^2$ expression. Doing so directly, however, is formally wrong because you cannot apply a pseudo-Riemannian metric directly to points in that fashion. So in this case the question is doubly wrong: It *does* matter, in principle, where the indices are placed and you cannot even write something like $x_\mu$ for a set of coordinates.

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# Group indices

The usage of indices in group theory is completely different, and a priori there is no notion of "upper" or "lower" indices. Given a group $G$ and a representation $\rho : G \to \mathrm{GL}(V)$ of some vector space $V$, one can of course choose a basis $v_i$ of $V$ and write any group element as a matrix $\rho(g)_{ij}$.

The notion of upper and lower indices enters here for groups where all or most of the irreducible representations can be constructed from tensor products of the fundamental representation: One declares vectors in the fundamental representation to have components with indices $v_i$ and those in the conjugate fundamental representation to have components with $v^i$ (or vice versa) and then one can write $T^{\mu_1\dots\mu_m}_{\nu_1\dots\nu_n}$ to denote an element of $\bar{V}^{\otimes m}\otimes V^{\otimes n}$. This shorthand is useful to then deduce which combination of indices and their (anti-)symmetrization correspond to irreducible representations, see e.g. [this answer](https://physics.stackexchange.com/a/14586/50583).

Again, the upper and lower indices are related, but do not denote the same objects, and they signal a different transformation behaviour under the group (fundamental vectors transform by $\rho(g)$ while anti-fundamental vectors transform by $\bar{\rho(g)}$), just like indices in the geometric case signal different transformation behaviour under coordinate changes.