Of course the Sun can be in the East and the West at the same time!!! It depends on where the observers are standing.
Imagine you are looking down on the Earth from a million miles above the North pole, and your buddy is looking down from a million miles above the South pole. And it's noon in London, so the Greenwich meridian is directly facing the Sun. The half of the Earth facing the Sun is lit, and the other half is dark.
If Linda is in London, the Sun is overhead. If Zack is in New Zealand, the Sun is out of sight behind the earth (under Zack's feet). If Imogen is in India, the Sun is just setting in HER West, and if Carl is in California, the very SAME Sun is just rising in HIS East. From your vantage point above the North pole, you can see all those observers at once. (So can your buddy, except that you see them L-I-Z-C anticlockwise, and he sees them L-I-Z-C clockwise.) That's why we have time zones. But the point is, East/West are earth-relative, and day/night are Sun-relative, and the Earth rolls round its axis in the warm sunlight at 15 degrees per hour.
The Moon is also exactly half-illuminated all the time. When it is on the half of its orbit on the Sun side of earth, most of the illuminated bit is hidden from us by the moon itself, and everybody on earth gets rotated past the visible crescent bit every 24.8 hours (because the moon advances in its orbit a little every day); and they see that crescent in daylight because the moon is on the same side of the earth as the sun is. At some stage the crescent becomes a sliver, nothing, and then a new crescent on the other side.
Conversely, when the moon is in the part of its orbit nearer the outer side of the earth, we can see the part of the moon that is completely illuminated, so it is close to being full, but we can only see that state from the dark (i.e. night) side of earth.
Get hold of a torch (as the Sun), an orange (draw on a few countries and the poles), and a small potato. Put the moon in various phases of its orbit, shine torch sunlight on them both, then for each phase, rotate the earth and visualise what you can see from each country on it during a day.
I mentioned how the moonlight shines on the snow at home in Scotland, to a guy I met when I was working in California, and he said "Hey, have you got a moon there, too?".
What messes my head is that, in the southern hemisphere, the Sun rises in the East and goes round the North of you during the day before it sets in the West. And you have Christmas in June, of course.