It is correct to assume photons emitted by the Sun are the whole electromagnetic spectrum, excluding gamma rays?

That used to be the prevalent thinking.  In contrast, the current thinking is that the entire electromagnetic spectrum is emitted by photons from the Sun. 

The electromagnetic spectrum starts with the longest waves, which are radio waves.  Next up are microwaves, which are a little shorter than radio waves.  Then, infrared, or heat waves, which are even shorter in wavelength.  The visible light spectrum comes next, followed by ultraviolet and X-rays.  Gamma rays used to be thought of as the product of the whole universe, but more modern thinking includes gamma rays as the product of photons from the Sun as well.  It is thought cosmic ray electrons collide with photons from the Sun, creating a large gamma ray laden light show for the inner solar system.  So all the lower end of the electromagnetic spectrum, and now, the upper end as well are believed to be emitted by the Sun: Gamma rays are no longer excluded from the concept of what parts of the electromagnetic spectrum are emitted by the Sun.