have discovered rodent droppings in my house (ugh!) Do the ultrasonic repellants work? How do exterminators actually catch the animals inside the house?

Some times they use dangerous fumes of poison to kill these rodents.
Most of the time baiting is the best way to go, you can use baits,traps or glue boards to get rid of rats or mice. You just have to remember that it could take more then one baiting to do the job correctly. I think that fumigation of some sort would be the last resort. I have been in this business for a long time and never had to resort to fumigation for the exterminating of rodents or mice.