Hi Tor,
The detail of the theory explains both of these problems. Indeed I believe that the Earth and Moon have similar surface compositions is one of the key lines of evidence for the theory.
First, this happened during the formation of the solar system, when there may have been more planets than we find in the solar system today. The proto-Earth and its impactor, another protoplanet probably of similar size to Mars, would have been in similar orbits. The planets would interact gravitationally which could shift them into different orbits, eject one of the protoplanets out of the solar system, or as in this theory the protoplanets collided. The merged planet and moon would follow a similar orbit around the Sun to the original protoplanets. It is worth noting this orbit isn’t necessarily exactly where the Earth and Moon are found today as gravitational interactions between the planets were common in the early solar system, before the planets settled into the orbits we find today.
In the massive impact, the outer surface of the proto-Earth and the impactor would get mixed together. Some of this material would be ejected into space, settle into an orbit around the Earth and coalesce to form the Moon. Hence the surface of the Earth and Moon are similar compositions, a blend of the proto-Earth’s surface and the impactor. You can get a better idea from this video.
https://www.youtube.com/watch?v=BoU9OA92trA
Best wishes,
Andy