Do tools exist which automatically find copy-and-paste code?

Our CloneDR is a tool for finding exact and near-miss blocks of code constructed by copy and paste activities. It can handle systems of millions of lines of code.

It uses precise language grammars to pick out language structures (identifiers, expressions, statements, blocks, functions, classes, packages, ...) that have been copied, and to determine the points of variation across the sets of clones (any of those structures can be parameters!)

CloneDR operates on a wide variety of languages: C, C++, C#, Java, PHP, COBOL, Python, Ada, Fortran, EGL and visual basic (VBScript, VB6, VB.net).

The website has a number of sample clone detection reports from a variety of those languages.

This product is available for evaluation on http://www.semanticdesigns.com. Other open source alternatives are Simian and PMD CPD

Yes, try the Copy Paste Detector.

http://patterninsight.com/products/cp-miner.php
Related paper - http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.123.113