Raising and lowering indices in line elements - why do we raise and lower them in line elements?

In general, a $(p,q)$-tensor eats $p$ covectors and $q$ vectors and returns a real number; such an object has $p$ indices upstairs and $q$ indices downstairs. The components of the metric tensor are written $g_{ij}$ because the metric $mathbf g$ is a $(0,2)$-tensor which eats two vectors (and no covectors) and returns their inner product.

If you see the indices written upstairs, then you're not talking about the components of the metric - you're talking about the components of the dual metric $tilde{mathbf g}$, which defines an inner product on the space of covectors. The components of the dual metric are the matrix inverse of the components of the metric, which means $$tilde g^{ij} g_{jk} = delta^i_k$$ Conventionally, we drop the tilde and write the components of this object simply as $g^{ij}$, distinguishing it from the metric itself only via placement of the indices.

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In Euclidean space and cartesian coordinates, the components of the metric $g_{ij}$ written out in array form are

$$g_{ij} = pmatrix{1 & 0 &0 &0&1&0&0&0&1&0&0&0&1} equiv delta_{ij}$$

My first question is simply this: why do we write $delta_{ij}$ here, when I am used to seeing the Kronecker delta written as $delta^i_j$, with one raised index?

Those two symbols have subtly different meanings. They are both equal to $1$ when $i=j$ and $0$ otherwise, but the former transform as the components of a $(0,2)$-tensor (which is to be expected, since it is the metric tensor expressed in a particular choice of coordinates) while the latter transform as the components of a $(1,1)$-tensor (or a linear transformation).

In other aspects of my work, involving quantum field theory, the convention is to write the standard Minkowski metric as $eta^{ij}$, with two raised indices, as opposed to two lowered indices.

As above, that is the Minkowski dual metric, which is a $(2,0)$-tensor. In cartesian coordinates, the components $eta^{ij}$ are the same as the components of $eta_{ij}$, but the same is not true e.g. in spherical coordinates, in which case

$$eta_{ij} = pmatrix{-1 &0&0&0&1&0&0&0&r^2&0&0&0&r^2sin^2(theta)} qquad eta^{ij} = pmatrix{-1 &0&0&0&1&0&0&0&frac{1}{r^2}&0&0&0&frac{1}{r^2sin^2(theta)}}$$

Why, in this formula, have we suddenly gone from writing lengths such as $dx^i$ with raised indices to ones such as $x_i$ with lowered ones?

If you go back a step, we have

$$mathrm ds^2 = |mathrm dmathbf x|^2 + K frac{(mathbf xcdot mathrm dmathbf x)^2}{a^2-K|mathbf x|^2}$$ $$ = delta_{ij} mathrm dx^i mathrm dx^j + K frac{(delta_{ij} x^i mathrm dx^j)(delta_{ell m} x^ell mathrm dx^m)}{a^2 - K|mathbf x|^2}$$

Relabeling dummy indices on the second term, we have

$$mathrm dx^2 = left(delta_{ij} + K frac{(delta_{iell} x^ell mathrm )(delta_{jm} x^m mathrm )}{a^2 - K|mathbf x|^2}right) mathrm dx^i mathrm dx^j$$

If we use the index-lowering convention $x_i equiv delta_{iell}x^ell$, then comparing this expression to the general $mathrm ds^2 = g_{ij}mathrm dx^i mathrm dx^j$ yields the components given in the text. As a side note, it is my personal belief that using the index-lowering convention on the coordinates is not a good idea. It is, however, perfectly well-defined shorthand, so your mileage may vary.