Do all stars of a given age and mass have relatively the same emission spectrum? My understanding is that much of the visible light that we see during the day is given off from the Sun, and that means that most of the colors that we see are dictated by (with the exception of atmospheric effects) the particular wavelengths of color that the Sun emits, which does not include the entire continuous color spectrum.

Here is an image of the wavelengths emitted by the Sun

My understanding is also that, at least during a particular part of the life cycle of a star, all stars are burning balls of hydrogen and helium. If that is the case, then do all stars at that point in their life give off emission spectra of visible light that is similar to our own? Is it possible that an observer on a planet with an atmosphere similar to that of Earth might see different ranges of color than we do, simply due to, perhaps, the composition of their star? Could there be planets similar to Earth where the sky would appear purple rather than blue?

Let me know if any of my assumptions are incorrect.
The spectrum of light emitted by a star depends mainly on its surface temperature, with smaller variations associated with composition, surface gravity and magnetic activity$^{1}$.

The surface temperature of a star is determined by its mass and evolutionary phase and therefore by its age.

Stars spend most of their lives on the core hydrogen burning main sequence. During this phase, a simple rule is that more massive stars have hotter surface temperatures and their spectra are more heavily weighted to shorter (bluer) wavelengths.

The appearance of the sky is caused both by the spectrum of our Sun and the physiology of the eye. Light is scattered in our atmosphere according to the inverse fourth power of wavelength. At the same time, there is comparatively less violet light in the spectrum of sunlight at the top of the atmosphere and the eye is less sensitive to that light (and some of it is absorbed, most of its ultraviolet wavelengths). It is the case that a hotter star than the Sun would have a spectrum richer in violet light and this would increase the amount of violet scattered light from the sky. Whether that would be sufficient for us to view the sky as purple, with our eye physiology, I am unsure; but probably not (see Why are there no green stars? ).

More massive stars also "evolve" faster, because they burn their available fuel much faster. An "evolved" star is one which has finished fusing hydrogen in its core and moved on to "shell" burning around the core or burning heavier fuels in its core. Shell burning stars tend to be bigger and cooler than main sequence stars of the same mass.

$^{1}$ Increased magnetic activity can drastically increase the amount of light from a star at UV and X-ray wavelengths, by orders of magnitude. This is primarily a feature of comparatively low-mass stars (like the Sun or lower) and is driven by rapid rotation. Rapidly rotating (single) stars are young. This extra short wavelength light is not energetically important to the star, in that the vast majority of its radiation is still at visible and infrared wavelengths.