I'm flying from Croatia to Poland via Stuttgart. Will I need to go through immigration in Stuttgart?
I have a 1-hour layover; is that a sufficient amount of time?
Yes immigration will be in Stuttgart.
I've never flown through Stuttgart, but I've made connections in larger Schengen-area airports (Munich, Frankfurt, Vienna, Athens) with a similar amount of time, and sometimes even less. I wouldn't worry too much about making the connection.
In any event, if you miss the connection, the airline will accommodate you on another flight.