is this the proper identification of this cyclic molecule?
It's correct, as already mentioned in a comment.

Some might find more or less helpful to redraw the structure to:



Your additional question can be tricky, understood a different way. It's possible to exchange H and Cl positions, or rather axial to equatorial bonds and the other way around – by switching to the other boat conformation, i.e. without changing the configuration:



(The two structures are identical, one of them can be rotated to superimpose the other.)