Does the sun always rise in the east?

Answer

You are right, in the summer the sun rises in the northeast, in the winter in the southeast. And it sets in the northwest in summer, and in the southwest in winter.
Here in Flanders, for example, the sun rises on June 21, at an azimuth of about 50°. The azimuth is measured along the horizon, starting from the north (azimuth = 0°) over the east (90°). So you see that the sun rises then, 40° away from the east to the north. In the evening, the sun sets that day at an azimuth of 310°, while the west is at 270°. In winter on December 21, we find the same deviation of 40° from east and west, but in the direction of the south.
Incidentally, if your friend were right, each day would have exactly 12 hours of light, and 12 hours of night. At our northern latitude we have 16 hours of light and 8 hours of night on June 21, and on December 21 this is the other way around.

The further you move from the equator, the more extreme this becomes. It becomes so extreme that in the summer the sun near the North Pole never even sets, and in the winter it never rises. This is reversed at the South Pole. If your friend were right in his statement that in a certain place the sun always rises and sets at the same place on the horizon, how could he explain the latter (the midsummer sun)?

At the equator, the sun does rise in the east, and sets in the west, along an arc that passes through the zenith on March 21 and September 21, deviating from it in summer or winter, respectively to the north or to the south.
(for the sake of simplicity, I assume that the seasons start exactly on the 21st of those months)