Question 1: If we assume that the center of mass coincides with the origin of coordinates and is centered on the Sun, we know that coordinates of Sun-Earth-$L_4$ and Sun-Earth-$L_5$ are the vertices of equilateral triangles(Proved by Lagrange 1772). So the angle between the line joining the Sun and Earth and the line joining Earth and $L_4$ or $L_5$ are $\pm 60\deg$. Measuring the position of $L_4$ from the Sun gives $(r\cos(60),r\sin(60),0)$ and $L_5$ gives $(r\cos(60),-r\sin(60),0)$. Notice there is no difference in this case between measuring from Earth or the Sun. The numerical value of the $L_4$ and $L_5$ coordinates are the same in either case resulting from the fact that they are vertices of an equilateral triangle. The x-coordinate splits the distance between the Earth and Sun in half. You can see this by rotating a Sun centered frame of reference at the orbital frequency of the earth or alternatively by rotating an Earth centered coordinate system at the same frequency. Everything of interest in this problem is stationary in these rotating frames.
Question 2: Care must be taken now on the location of the origin of coordinates. Assuming we measure coordinates from the Earth, Locating the object is accomplished by drawing 2 lines and finding their intersection. These lines emanate from $L_4$ and $L_5$ and pass through the object. Let the Earth be located at point $E$, and the object at point $O$. Initially the problem states that $\angle OL_4E = 55\deg$ and $\angle OL_5E = 45\deg$. Now use the properties of equilateral triangles, some Euclidean geometry and the radius of the circular orbit to calculate the coordinates of the object. Hope this helps.