Lowering and Raising Kronecker Delta When an index of the Kronecker-delta tensor $\delta_a^b$ is lowered or raised with the metric tensor $g_{ab}$, i.e. $g_{ab}\delta^b_c$ or $g^{ab}\delta_b^c$, is the result another Kronecker-delta tensor?
 A: If you are dealing with spacetime indices (i.e. tensors over the spacetime), then symbols like $\delta^{ab}$ or $\delta_{ab}$ don't make sense. If you lower an index of $\delta^a_b$ you will end up with the metric $g_{ab}$, same for raising an index. This is clear from the definition of $\delta$:
$$g_{ab}\delta^b_c=g_{ac}$$
and
$$g^{ab}\delta_b^c=g^{ac}$$
since $\delta^b_c=1$ for $b=c$ and $\delta^b_c=0$ for $b\neq c$.
A: One answer is that the delta operates on the metric tensor, changing its (the metric tensor's) index (one of its indices).
Another answer is that the metric tensor operates on the delta, lowering one index.
Both answers must be correct and thus they are equal.
So you get delta with two bottom indices = g with two bottom indices.
The insight is that the delta tensor and the metric tensor are one and the same entity. When both indices are covariant or both are contravariant, we usually (always, in practice) use the symbol g; when one is covariant and the other is contravariant, we use the symbol delta.
When it's g, we think of it as the metric (or an index lowerer or raiser). When its delta we think of it as the identity (or as an index changer). But they are different aspects of the same thing.
For practical purposes, we think of them as different, but really, strictly, they're actually, ultimately the same tensor.
So it's not quite right to say there's no such thing as delta with two lower (or two upper) indices. It's just that we always write it as g.
Similarly, what is g with one lower and one upper index? Answer: it is delta with one lower and one upper index.
g is delta in disguise and delta is g in disguise. 
A: The result is the metric: the effect of the Kronecker delta in your examples is to set $b = c$. The Kronecker delta is really just the identity matrix.
A: You're confusing two things. In tensor calculus, the Kronecker delta should be visualized as basically the identity. What it does is relabel an index. Example:
$$g_{ab}\delta^b_c=g_{ac}$$
This has nothing whatsoever to do with the Dirac delta function (it's actually a distribution) in this context. In nonrelativistic quantum theory Dirac delta distributions might be used e.g. for orthogonality relations between eigenstates, just like the Kronecker delta, but this, in turn, has nothing to do with tensor calculus.
EDIT: This is an answer to your original question, which has since then been edited by Qmechanic, quite drastically changing the second part. I'm not sure this edit accurately reflects your confusion. If it does, please ignore the second part to my answer.
