Attached my solution.

Actually Roberto, there's a very minor error in your posted solution:

dx/ds is not equal to v, it's proportional up to some scalar function of s, which depends on the choice of parameter s.

However, even with v = f(s) . dx/ds (where f(s) is a scalar), your first and third terms still cancel, since the f(s) appears in both - so nothing else in the solution is actually affected by this.

Yes, of course you are right.
In my calculations I made the standard choice for s: the affine parameter, i.e. the proper time. This makes v equal, instead that just proportional, to dx/ds. But I omitted to make explicit my choice; furthermore, this choice is not really necessary, as you pointed out: it is just a matter of convenience.