Are there any benefits/disadvantages/traits from having a permanent lower core temperature?
Yes absolutely. Body temperature is associated with basal metabolic rate which is linked to life-span/body mass. Keeping your body at temperature is responsible for the majority of your caloric burden majority of your caloric burden, and having a lower resting temperature would mean you require fewer calories to survive. Higher body temps are thought to have been a defense against parasite infection, so reductions in body temperature might leave you more susceptible to disease, however a paper came out recently discussing falling body temperature in the US and its implications, and I think it addresses many of your questions in detail in the discussion.
Edit: To elaborate on that elife paper, they've identified a small reduction in average body temperature overtime using military medical records going back to the civil war. They postulate that the reductions they observe are due to improvements in healthcare and a potential decrease in chronic inflammation in the population which increases your body temperature.