Shift Vector in Warp Equation

I believe the indices are raised and lowered using the spatial $gamma_{ij}$ as metric.

This is how the dot product can be defined for covariant and contravariant vectors (without explicitly inserting the metric) i.e, with the metric, the dot product would look like $$beta cdot beta = beta_i beta^i = gamma_{ij} beta^j beta^i$$

Note that $$beta_i = gamma_{ij} beta^j $$ and $$beta^i = gamma^{ij}beta_{j}$$

The metric $gamma_{ij}$ raises and lowers indices in the equation you have mentioned.

The sub and superscripts represent contra and covariant vectors. We can (sometimes) think of these vectors as column vectors (contravariant) and row vectors (covariant or covectors). An intuitive way to understand the difference between the two is to consider how each transforms under a change of basis. A covector transforms in the same way as its basis transforms, whereas a contravariant vector transforms inversely to how its basis transforms. An example would be the change of units from m to cm on a contravariant position vector, doing that transformation multiples all components of the position vector by 100 whereas the basis is changing by a factor of 1/100. A covariant position vector (sometimes called a dual vector) would have its components divided by 100 in a change of units from m to cm just like its basis. To your question, the contra and covariant indices on the shift vector imply a sum over spatial coordinates only since they are latin scripts and not Greek spacetime indices, doing so effectively gives the magnitude of the given component for a given spatial coordinate. As to its more physical meaning, I can't help there sadly.