Differential Geometry Notation When we have a metric on a manifold, there is a natural isomorphism between the tangent space and the cotangent space, and so, if I understand correctly, it is not so important to keep track of which indices are up and which are down since you basically always implicitly assume that if an index that was defined to be up is down the metric has been used to "lower" it, and vice versa. 
Thus, when we do have a metric, why do we ever need to care about which indices are up and which are down? Couldn't we, by convention, write all indices down (as is done in the rest of mathematics) and finish with this story?
 A: No, that is, if you do not want to have the derivatives of the metric tensor crop up everywhere. 
You could also do differential geometry essentially without indices and insert the basis and dual basis at appropriate places. If you use the tradition of latin letters for the primal and greek letters for the dual basis, then lower indices everywhere are acceptable. But the formulas get much longer.
A: It might be helpful to consider an alternative system of doing differential geometry and how it handles this problem.
"Gauge Theory Gravity" was developed to use clifford algebra and "frame fields" on an intrinsically flat manifold for gravity, but it is essentially a framework for differential geometry.  It replicates the usual constructs of differential geometry using a linear map $h$, called the "displacement (or position) gauge field" that is kinda like the square root of the metric.  In particular, $g(a, b) \equiv h^{-1}(a) \cdot h^{-1}(b)$ for two vectors $a, b$.
You might notice I said vectors, and not covectors.  The elements of the base manifold aren't really distinguished from each other because it's a flat space, so in principle, you could put covectors in instead.  The distinguishing between vectors and covectors comes in how $h$ is used to develop a geometrically significant quantity (i.e. one that obeys tensor transformation laws).  Vectors always use $h^{-1}$, and covectors always use $h^T$.  Derivatives, like the covariant derivative $\nabla$, must always be wrapped in $h^T$ to get $h^T(\nabla)$ instead, while a curve $\alpha(t)$ will naturally pick up $h^{-1}$ when differentiated to get $h^{-1}(\alpha')$.
So while you could transform all covectors using the metric so that everything is wrapped in $h^{-1}$ instead of $h^T$, it's somewhat cumbersome to do so.  The GTG approach allows you to just define quantities like $\tilde v = h^{-1}(\alpha')$ and forget how the bare quantities were wrapped in the first place.  Dot products between quantities that obey the proper transformation laws are inherently meaningful from a geometric standpoint, so there's no issue.  This is something conventional differential geometry can't do!
So until DG comes up with something like that, you're kinda stuck.  You can write all formulas and such assuming only vector arguments (do Carmo does this in his Riemannian geometry book), or you can keep track of everything by hand.  I prefer the latter approach because while there is an equivalence, keeping track of whether a quantity is vector or covector or such can tell you a bit about the nature of that quantity: did it come from a gradient, say, or is it a derivative along a curve?
GTG may be a bit "out there," but if you're studying gravity, I could consider looking into frame fields at least (or tetrads, or vierbeins; they all mean the same thing).
