Most members of the animal kingdom can regenerate lost body parts at some (or all) of their life cycle. Mammals aren't one of those animals. Yet, scientists say the pathway (called the Wnt pathway) still exists, untapped, in mammalians.
Before we start speculating on alien sentient beings being able to routinely sprout new appendages (or like Jeebs, the hapless alien in MIB I and II, a new head), the question that first needs to be pondered is why have mammalians lost, or suppressed, this ability. On the surface, it seems like it would be evolutionary advantageous to retain this ability. Yet apparently it was also advantageous to lose it. Even among creatures that can regenerate, many of them can not do so when adults. This seems to indicate that there must be some problems with regeneration, especially with advanced life forms.
It is obvious we do not fully understand all the parameters regarding regeneration. However, the fact remains the Wnt pathway still exists, latently, but exists in at least some higher mammals. Thus, maybe it is not far-fetched to think there could be sentient aliens with at least some regenerative abilities. In addition, since Earth scientists are looking to restart this latent ability (and have had some success with adult frogs and chicken embryos), it is a distinct possibility that other alien sentient beings could work on restarting any latent regeneration they may have, and may be successful, either as medical procedure, or as a full time reacquired ability.
However, the more advanced the creature, the more there are limitations to the regeneration - that only makes sense. Simple structures are easier to rebuild, with less chance for mistakes. Not so for complex structures. Simple structures require less energy to rebuild than complex structures. Simple structures can rebuild faster. And finally, simple creatures tend to less centralized, the less centralized they are, the easier they can live without a part of themselves. Complex creatures tend to have rather centralized controls. Which brings us to an important point: there are structures that just can't be regenerated - such as a heart, or brain. How can a mammalian body naturally survive a missing heart even if it has the ability to regenerate? The body cannot survive long enough without a heart to give time enough for another heart to regenerate. Same with a brain - if the entire brain is dead or missing, the body cannot continue functioning long enough to allow for regeneration to complete (and even if a creature could regenerate a new head, ala Jeebs in MIB, how could it retain all its knowledge and memories?).
So, how would this affect culture, if we could routinely regenerate missing ears, noses, arms, or legs? For one, it would prolong life - for instance, amputations due to illness would be a temporary set back as the body regrows a healthy new appendage. Maybe more people would be greater risk takers, since the risk of death or permanent dismemberment or disfigurement is lowered (cosmetic surgery? Just remove the offending area, and let a new area grow back). But it also may make war or violence even more common place, since soldiers could be more easily resent back into battle (wars of attrition would take much longer). Of course repeated violence, repeated being torn apart and regrowing must have an affect on the brain, especially on the mental "adaptation" to such a "life." This would also affect culture.
The tougher question is how would it affect theology? What kind of theology would an alien race that had strong natural regenerative powers have?
Casselman, Anne. "How to Grow a New Limb." Discover. October 2007. 17.