Can I replace this circuit with an IC?
"Secondary question: how do you even know things like the ULN2803A exist?"


Search Google for something like "transistor array".  Add words like "IC" or "semiconductor" to weed out unrelated stuff.

http://www.google.com/search?q=transistor+array+semiconductor

finds

http://www.onsemi.com/PowerSolutions/product.do?id=ULQ2003ADR2G

When you find a manufacturer's listing of a specific part that looks vaguely relevant, use the manufacturer's site navigation to "go up one directory" and see what similar parts they have.  That page has a breadcrumbs navigation at the top "Home >  Products >  Product Catalog >  Drivers >  Load / Relay Drivers >  ULN2003"

Click "Load / Relay Drivers" to see similar stuff.