Does the Morse homology depend on the orientation?

Yes, it's independent of these choices - if only because it turns out to be isomorphic to singular homology. The need for orientations is a necessity of working with $\Bbb Z$ coefficients instead of $\Bbb Z/2$ - when writing down the differential, we need to know when a flow line is "positively oriented" or "negatively oriented" to properly count them with signs. This leads us to the unseemly need for this choice.

One reason many books first work mod 2 (and sometimes Floer homology books will do the same, at least at first) is precisely so that one doesn't have to worry about these orientation issues. The ambiguity from the orientations is a bunch of $\pm 1$s induced in the differential - but in $\Bbb Z/2$, there is no ambiguity at all, since $1=-1$. Indeed, when working mod 2, we don't need to assign signs to each flow line - we just need to count them.

If you change the orientation on $W^s(a)$ then an isomorphism from the old complex to the new is given by sending the generator $a$ of $C_k$ to $-a$ (and acting as the identity on all other critical points).