Would it be possible to have a planet where the points the Sun rises and sets change every 5 months? Would life as we know it be possible there?
Yes. You’re living on one of them (assuming you’re on Earth - if you aren’t, please tell me what planet you are on, and how you’re getting internet access).
Because of the tilt of Earths axis of rotation, the point on the horizon the Sun rises or sets will change as the year goes on. In the northern hemisphere, this point of sunrise slowly drifts north until the summer solstice, then drift back south until the winter solstice. Here, the Sun’s rising point will start to drift north again.
For the southern hemisphere, reverse it.
The Earth’s axis tilt is 23.5 degrees. If you increase the tilt, the Sun will sweep a larger area of the horizon between the solstices.
In the extreme case of a planet with a 90 degree tilt, the sun will sweep from north pole to south pole and every point in between as the planet spins around the sun on it’s side.
Uranus has an axial tilt of 98 degrees.
Uranus is all messed up.
If you had an Earth-like planet with that degree of axial tilt, life would probably still be possible, but your days and nights and seasons would be all messed up.
For about three months, the northern hemisphere would be mostly in darkness while the southern was in light. For the next thee months, they would change places with days in the northern hemisphere getting longer and shorter in the southern hemisphere. this would continue until the southern hemisphere was in darkness and the northern bathed in light. Then, a reversing for the next three months until the norther hemisphere was once more in darkness.
Enjoy your really extreme seasons.