Does anyone know how the 2nd Punic War in particular would have affected Republican Rome culturally? The Carthaginians were known to have had a formidable culture, so did increased interactions between Romans and Carthaginians during the Punic Wars in any way influence Roman culture, specifically in terms of religion?
The Romans were quite nervous about the Carthaginians, particularly because they practiced infant sacrifice (something which the Romans deplored. Moreover, there is scholarly debate on this topic; I tend to believe the ancient authors). There were times where, especially during Hannibal's invasion, the Rome began to sacrifice to all sorts of gods, and perhaps some Carthaginian deities came in there. One of the biggest influences Punic culture had was with regard to foodways. Cato gives a recipe for a caserole called "Puls Punica", literally Punic porridge. Additionally, the Carthaginians were renowned for their fine and ornate baking. Terracotta baking ware has been discovered showing just how ornate they were. Roman bakers would have been quite keen on getting the recipes because they were exceedingly popular. I would recommend looking at this book for more info:
Charles-Picard, Gilbert, and Colette Charles-Picard. Daily Life in Carthage: At the Time of Hannibal; Transl. from the French by A.E. Foster. London: Allen Unwin, 1961. Print. Hope this helps!