In this subsection, "location" is meant in the general sense, i.e. not as point-like location only.
(Theorem) If l is a straight line in K parallel to v, and l' is the corresponding location in K' marked out at (coordinate) time t in K, then l' is a straight line parallel to v'.
(Proof) Viewed from K', any point P of l is moving at the constant velocity v'. On the other hand, in K all points of l', and only those from K', go through P. Thus, during the course of its movement in K', P meets exactly the points of l'. So l' is a straight line parallel to v'.
(Theorem) The location l' marked out in K' is the same for every t, and l' corresponds to l at any time t' in K'.
(Proof) The points of K' marked out at any t1 in K, and only those, are moving along l in K, so marking out at any other t2 in K results in the exact same points of K'. Now, switching the roles of K and K', we get that for every t' in K', l' corresponds in K to the very same straight line ll' parallel to v. But from the way l' was constructed we also know that ll' and l have common points, so ll' must be equal to l, since both of them are parallel to v.
The x'-axis of K' will always be chosen such that it coincides with the x-axis of K, at all times. To obtain the x'-axis, all we have to do is to mark out the location in K' that corresponds to the x-axis at any given time t in K. As for the orientation of the x'-axis, we define the order of its points, and with that the positive and the negative direction as well, by the order of the corresponding points on the x-axis at t. Independently of our choice of t, we'll get the same x'-axis and the same order of its points.
(Theorem) In K', v' points in the negative direction of the x'-axis.
(Proof) In K, a point P of the x-axis meets the points of the x'-axis in the above defined negative direction. If we imagine there is a particle at P, then causality implies that P, while moving at velocity v', meets the points of the x'-axis in the same order in K' (and so the above defined "directions" are proper directions in K' too).
For completeness, one last theorem about the x'-axis:
(Theorem) If in K', point P2' comes after P1' on the x'-axis, then at any given t', the corresponding point P2 comes after P1 on the x-axis of K.
(Proof) Both P1 and P2 are moving in the negative direction along the x'-axis in K'. At t', P1 is already at P1', so P2' must have met P1 before t'. Therefore, P2' meets in K' first P1 then P2. Due to causality, the order is the same in K. And since P2' is moving in the positive direction along the x-axis in K, P2 comes after P1.