Shotguns fine , 12 bore great for all your shooting , personally I use either a o/u or mostly my semi-auto , with regard to cartridge for all my shooting, game ,rough shooting , pest control etc , I use clear pigeon 32g no 6.
use what suits you and your chosen gun though.
For dear little bunnies I will use one of my shotguns , the rimmie or my pcp , controlling rabbits takes different methods for different situations.