Does an echocardiogram check for heart rhythm problems as well as structural problems, or is it limited to just picture view/structural?

Both: It tells you both structure and rythm, but it meant to evaluate for structure, measuring ejection fraction in each heart beat and evaluate pressure gradient in all four chambers. Ekg tells you rhythm.

Yes and No: An echocardiogram is an ultrasound of the heart, that specifically evaluates heart structure and function. The heart rhythm is monitored during the study to see heart rate, but this is not a test to specifically evaluate rhythm. The best test for that is an ekg (electrocardiogram) or holter monitor, which you wear for 1-2 days to monitor rhythm.