1. My understanding is that prime lenses generally contain fewer elements than zoom lenses. Hence, flare resulting from internal reflections is reduced for the former compared to the latter.
2. Stopping down will reduce the size of flare, where it exists, though I assume from one of your photos with a 'star' shaped sun that you are already doing that. If not, this point may be your best option before springing for a new lens. Certainly worth experimenting with - a flare-free image suffering from diffraction is better than the same image ruined by flare.
3. Using filters when shooting into the sun is, in my experience, a recipe for disaster due to reflections off of the filter.