1. What would you do with the child?

It's interesting as things have changed over the last 10 years. In the past there was concern that button batteries may corrode, or that they might set ujp small electrical currents that may encourage mucosal erosion and even perforation. Fortunately there is evidence out there to help us.

An analysis of over 2000 cases suggested that those batteries that pass into the stomach can be safely left.
http://pediatrics.aappublications.org/cgi/content/abstract/89/4/747

An X-ray will show you where the battery is, or rather were it is not!

In general if the patient is well and the batteries are demonstrably beyond the oesophagus then expectant management is indicated.

The use of regular radiology to confirm passage is controversial as many of us do not advocate irradiating children unless absolutely necessary. It is more common practice in US departments.
http://www.emedicine.com/emerg/topic139.htm